PREDICTION OF COHESIVE SEDIMENT MOyEKEMT IN ESTUARIAL WATERS By EARL JOSEPH HAYTER DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVElSTTf OF PWRIDA IN PARTIAL FULFILLMENT OF THE REOUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1983 ACKNOWLEDGEMENTS The author would like to express his sincerest appreciation to his research advisor and supervisory committee co-chairman. Or. A.J. Mehta, Associate Professor of Coastal and Oceanographic Engineering, for his continuous guidance and encouragement throughout the course of this research. Appreciation is also extended for the valuable advise and suggestions of the supervisory committee chairman. Dr. B.A. Christensen, Professor of Civil Engineering, as well as for the guidance received from the other committee members: Dr. B.A. Benedict, Professor of Civil Engineering; and Dr. J.L. Eades, Associate Professor of Geology. Sincere thanks are also due to Dr. E. Partheniades, Professor of Engineering Sciences, whose interest and suggestions were of a great help to the author. The author wishes to acknowledge the assistance provided by Dr. Ranjan Ariathurai to this investigation. He provided a copy of the finite element solution routine used in this study and gave advice regarding its use and related experimental research. Appreciation is extended to Drs. D.G. Bloomquist and P. Nielsen for their suggestions and providing references on various aspects of this study. Special thanks go to Mr. Vernon Sparkman and the staff of the Coastal Engineering Laboratory for their assistance with the experiments ii performed during this research. The author wishes to thank Ms. Debbie LaMar for typing this manuscript and Ms. Lillean Pieter for drafting the figures. In addition, the author thanks Ms. Lucile Lehmann and Ms. Helen Twedell of the Coastal Engineering Archives for their assistance. Great appreciation is extended to the Water Resources Division of the U.S. Geological Survey for their financial support of this research through the thesis support program. More specifically, the author expresses his gratitude to Dr. Robert Baker, regional research hydrologist, and to Drs. Carl Goodwin and Harvey Jobson who served as thesis support advisors. Finally, the author thanks his wife Janet for her love, moral encouragement and patience, and his parents, George and Lois Hayter, for their love and support. iii TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS ii LIST OF TABLES viii LIST OF FIGURES ix ABSTRACT xvii CHAPTER I INTRODUCTION . 1 1.1. Estuarial Cohesive Sediment Dynamics 1 1.2. Sediment Related Problems in Estuaries 4 1.3. Approach to the Problems 7 1.4. Scope of Investigation 9 II BACKGROUND MATERIAL 12 2.1. Introductory Note 12 2.2. Description and Properties of Cohesive Sediments 12 2.2.1. Composition 12 2.2.2. Origin 13 2.2.3. Structure 14 2.2.4. Interparticle Forces 15 2.2.5. Cation Exchange Capacity .... 19 2.2.6. Coagulation 21 2.3. Significance of Important Physical Factors in Estuarial Sediment Transport 30 2.3.1. Estuarial Dynamics 30 2.3.2. Sediment Processes 35 III SEDIMENT TRANSPORT MECHANICS 47 3.1. Introductory Note 47 i V PAGE 3.2. Governing Equations 47 3.2.1. Coordinate System 47 3.2.2. Equations of Motion 47 3. 2.1. a. Continuity 49 3.2.1.b. Conservation of Momentum 49 3.2.3. Advection-Dispersion Equation 51 3.3. Sediment Bed 53 3.3.1. Bed Structure 53 3.3.2. Effect of Salinity on Bed Structure 64 3.3.3. Bed Schematization 71 3.4. Erosion 75 3.4.1. Previous Investigations 75 3.4.2. Effect of Salinity on Resuspension 92 3.4.3. Erosion Algorithm 101 3.5. Dispersive Transport 106 3.5.1. Dispersion Mechanisms 106 3.5.2. Dispersion Algorithm 112 3.6. Deposition 116 3.6.1. Previous Investigations 116 3.6.2. Effect of Salinity on Deposition 133 3.6.3. Deposition Rates 140 3.6.4. Deposition Algorithm 152 3.7. Consolidation 154 3.7.1. Description 154 3.7.2. Consolidation Algorithm 168 IV MODEL DEVELOPMENT 178 4.1. Introductory Note 178 4.2. Review of Previous Models •^^^ 4.3. Model Description 181 4.4. Finite Element Formulation 188 4.4.1. Introductory Note 188 4.4.2. Shape Functions 189 4.4.3. Galerkin Weighted Residual Method 195 4.4.4. Equation Solvers 201 4.5. Convergence and Stability 201 V PAGE V MODEL VERIFICATION AND APPLICATION 203 5.1. Introductory Note 203 5.2. Laboratory Experiments 203 5.2.1. Recirculating Flume Experiments 204 5. 2.1. a. Facilities 204 5.2. l.b. Instrumentation. . . . 207 5. 2. I.e. Procedure 217 5.2.1. d. Results 222 5.2.2. Rotating Annular Flume Experiment 227 5. 2. 2. a. Facilities 227 5.2.2.b. Instrumentation. . . , 230 5.2.2.C. Procedure 230 5.2.2.d. Results 233 5.2.3. Model Simulations 233 5.3. Simulation of WES Deposition Experiment 236 5.4. Discussion of Results , 238 5.5. Model Applications 246 5.6. Model Limitations 257 5.7. Model Applicability , 258 5.7.1. Water Quality Problems ..... 258 5.7.2. Sedimentation Management Problems 259 VI CONCLUSIONS AND RECOMMENDATIONS 260 6.1. Summary and Conclusions 260 6.2. Recommendations for Future Research 269 APPENDICES A Derivation of Advection-Di spersi on Equation 271 B Coefficient Matrices in the Element Matrix Differential Equation 286 vi PAGE C Computer Program . 289 C.l. Main Program 289 C.2. Subroutines 290 C.3. Flow Chart 296 C. 4. User's Manual „ 308 0 Data Collection and Analysis Programs „ 323 D. l. Field Data Collection Program , 323 D.2. Laboratory Sediment Testing Program 330 REFERENCES 335 BIOGRAPHICAL SKETCH 349 vi i LIST OF TABLES TABLE PAGE 2.1 Properties of Sediment Aggregates (after Krone, 1963) 41 3.1 Principle Factors Controlling Erosion of Saturated Cohesive Sediment Beds 77 3.2 Cation Concentrations in Processed Sodium Chloride and Standard Sea Salt. 95 3.3 Variation of Empirical Coefficients in the Relationship Between Pi^^^) and T^^ 171 4.1 Quadratic Shape Functions 192 4.2 Derivatives of Shape Functions 194 5.1 Chemical Composition of the Tap Water (after Dixit, 1982) 217 V i i i LIST OF FIGURES FIGURE PAGE 1.1 Schematic Representation of Transport and Shoaling Processes in the Mixing Zone of a Stratified Estuary (after Mehta and Hayter, 1931) 3 1.2 Interactions of Tidal and Estuarial Sediment Transport Processes (after Owen, 1977). ...... 8 2.1 Repulsive and Attractive Energy as a Function of Particle Separation at Three Electrolyte Concentrations (after van Olphen, 1963) 17 2.2 Net Interaction Energy as a Function of Particle Separation at High Electrolyte Concentration (after van Olphen, 1963) 17 2.3 Net Interaction Energy as a Function of Particle Separation at Intermediate Electrolyte Concentration (after van Olphen, 1963) 20 2.4 Net Interaction Energy as a Function of Particle Separation of Low Electrolyte Concentration (after van Olphen, 1963) 20 2.5 Comparison of the Collision Functions for Brownian, Shear and Differential Sedimentation Coagulation (after Hunt, 1980) 25 2.6 Variation of SAR with Salinity (Sea Salt Concentration) (after Ariathurai, 1974) 27 2.7 Coagulation-Dispersion Boundary Curves for (a) Montmorillonite, (b) II lite and (c) Kaolinite at Three pH Ranges (after Kandiah, 1974) 27 2.8 Monthly Salinity Distributions in the Cumbarjua Canal, Goa, India; Ebb; Flood (after Rao etal_., 1976) 31 2.9 Variation in Chloride Concentration in San Francisco Bay and Sacramento-San Jaoquin Delta- September 1955 (after Orlob et_al_., 1967) 32 ix FIGURE PAGE 2.10 Salinity of the Surface Waters of the Pamlico River Estuary as a Function of the Distance from the Railroad Bridge in Washington, D.C. (after Edzwald et al_., 1974) 33 2.11 Computed Longitudinal Salinity Profile in the Yangtze River Estuary as a Function of the Distance Downstream from Jiang Zhen Dong for Two River Discharges (after Huang et al . , 1980) 33 2.12 A Plot of Raw Viscometer Data Obtained from the U.S. Army Corps of Engineers Philadelphia District Sample (after Krone, 1963) 39 2.13 Time and Depth Variation of Suspended Sediment Concentration in Savannah River Estuary (after Krone, 1972) 45 2.14 Schematic Representation of the Physical States of Cohesive Sediment in Estuary Mixing Zone (after Mehta et_al_., 1982a) 46 3.1 Coordinate System 48 3.2 Measured Red Density Profiles for Thames Mud for Two Different Consolidation Times (after Owen, 1970) 57 3.3 Measured Bed Density Profiles for Avonmouth Mud for Two Different Initial Suspended Sediment Concentrations (after Owen, 1970) 58 3.4 Measured Bed Density Profiles for Avonmouth Mud for Different Bed Thicknesses (after Owen, 1970) 59 3.5 Dimension! ess Density Profiles of Mud Beds (after Thorn and Parsons, 1980) 59 3.5 Variation of Bed Density with Depth for Three Different Conditions of Flow-Deposited Beds (after Parchure, 1980) 61 3.7 Dimensionless Density Profile of Mersey Mud with Tdc = 2 hours (after Bain, 1981) 62 3.8 Dimensionless Bed Density Profile of a Mud Bed (after Thorn, 1981) . 62 3.9 Dimensionless Density Profiles for Kaolinite Beds with T^c = 2, 5, 11 and 24 hours 63 X FIGURE PAGE 3.10 Dimensionless Density Profiles for Kaolinite Beds with T^^, = 48, 72, 96, 144 and 240 hours ... 63 3.11 Bed Shear Strength Profiles for Kaolinite Beds (after Parchure, 1980) 65 3.12 Bed Shear Strength Profile for a Kaolinite Bed (after Dixit, 1982) 66 3.13 Dimensionless Bed Density Profiles for Salinities of 0, 1, 2, 5 and 10 ppt 69 3.14 Bed Shear Strength Profiles as Functions of Salinity 70 3.15 Bed Schematization Used in Bed • Formation Algorithm 72 3.16 Hypothetical Shear Strength Profile Illustrating Determination of Bed Layers Thicknesses 72 3.17 Laboratory Determined Relationship Between Erosion Rate, e, and Bed Shear Stress, t:^ (after Mehta, 1981) 80 3.18 Example of Relationship Between £ and Stress Stress (after Mehta, 1981) 80 3.19 £ - ^^^^ °f Partheniades (1962), Series I and II (after Mehta, 1981) 81 3.20 Dimensionless ^ - \ Relationship Based on Results of Ariathurai and Arulanandan (1978) (after Mehta, 1981) 83 3.21 Relative Suspended Sediment Concentration Versus Time for a Stratified Bed (after Mehta and Partheniades, 1979) 83 3.22 Schematic Representation of the Selected Methodology for the Variation of the Applied Bed Shear Stress During Bed Preparation and Resuspension Tests (after Mehta et al . , 1982a) 86 3.23 Variation of Suspension Concentration with Time for Tj^ = 48 Hours (after Dixit, 1982) 87 3.24 C(T^') Versus t^- for Three Values of Tj^,, Using Kaolinite in Salt Water (after Mehta et_al_., 1982a) 88 xi FIGURE PAGE 3.25 Normalized Rate of Erosion, e^-/£ Versus Normalized Excess Shear Stress, (zi^) )/ T (zu). Using Kaolinite in Tap Water (after Mehta et_ al_., 1982a) 90 3.25 Normalized Rate of Erosion, Versus Normalized Excess Shear Stress, ^'^i~'^A^h'^^ ^ ^^(zij). Using Kaolinite in Salt Water (after Mehta et al_. , 1982a) 91 3.27 Critical Shear Stress Versus SAR for Montmorillonitic Soil (after Alizadeh, 1974) 94 3.28 Resuspension Rate Versus Normalized Excess Shear Stress 97 3.29 Slope, a , Versus Depth Below Bed Surface, z^, as a Function of Salinity 98 3.30 Ordinate Intercept, e^. Versus Depth Below Bed Surface, z^, as a Function of Salinity 98 3.31 The Internal Circulation Driven by the River Discharge in a Partially Stratified Estuary, (a) A Transverse Section, (b) A Vertical Section (after Fischer et_ al_., 1979) 110 3.32 Illustration of Wind-induced Circulation (after Fischer, 1972) 110 3.33 Ratio C/Cg Versus Time t for Kaolinite in Distilled Water (after Mehta and Partheniades, 1975) 120 3.34 Ratio C /C^ Versus Bed Shear Stress v (after Mehta and Partheniades, 1975) 120 3.35 Relative Steady State Concentration C^g in Percent Against Bed Shear Stress Parameter \-i (after Mehta and Partheniades, 1975) 122 3.36 C in Percent Versus t/t5Q for Kaolinite in Distilled Water (after Mehta and Partheniades, 1975) 124 3.37 Log t^q Versus for Kaolinite in Distilled Water (after Mehta and Partheniades, 1975) 126 3.38 ^2 Versus for Kaolinite in Distilled Water (after Mehta and Partheniades, 1975) 126 xi i FIGURE PAGE 3.39 Settling Velocity, W^, Versus Suspended Sediment Concentration, C, for San Francisco Bay Mud (after Krone, 1962) 127 3.40 Settling Velocity, W^, Versus Suspended Sediment Concentration, C, for Yangtze River Estuary Mud (after Huang etal_., 1980) 129 3.41 Versus C for Severn Estuary Mud (after Thorn, 1981) 129 3.42 Effect of Size and Settling Velocity of Elementary Particles on the Coagulation Factor of Natural Muds (after Bellessort, 1973) 132 3.43 Effect of Salinity on Settling Velocity of San Francisco Bay Mud (after Krone, 1962) 134 3.44 Effect of Salinity on Settling Velocity of Avonmouth Mud (after Owen, 1970) 136 3.45 Effect of Salinity and Suspension Concentration on Settling Velocity of Avonmouth Mud (after Owen, 1970) 137 3.46 Ratio C/Cg Versus Time as a Function of the Bed Shear, -c,^, for Lake Francis Sediment with 3=5 ppt 139 3.47 Ratio C /C^ Versus t. for Deposition Tests with LaRe Francis Sediment 141 3.48 Apparent Settling Velocity Description in Domains Defined by Suspended Sediment Concentration and Bed Shear Stress 141 3.49 Effect of Salinity and Bed Shear Stress on Settling Velocity of Lake Francis Sediment 146 3.50 Settling Velocity Versus Suspension Concentration for Deposition Test with Lake Francis Sediment. . . ISO 3.51 Variation of 0*^ with Salinity and 150 3.52 Variation of Mean Bed Density with Consolidation Time (after Dixit, 1982) 157 3.53 Variation of p/p^ ^^-^h Consolidation Time (after Dixit, 1982) 157 xi ii FIGURE PAGE 3.54 z^/W Versus p/p for Avonmouth, Brisbane, Grangemouth and Belewan Muds (after Dixit, 1982) 158 3.55 Zu/H Versus p/p for Consolidation Times (a) Less TFian 48 Hours and (b) Greater Than 48 Hours (after Dixit, 1982) 159 3.56 Normalized Bed Density Profiles for Thames Mud for Two Different Consolidation Times 160 3.57 Normalized Bed Density Profiles for Avonmouth Mud as a Function of Salinity 160 3.58 Normalized Bed Density Profiles for Avonmouth Mud for Different Bed Thicknesses 161 3.59 Variation of ■^r^zu) with Zj^ for Various Consolidation Periods (after Dixit, 1982) 165 3.60 Correlation of Bed Shear Strength with Bed Density (after Owen, 1970) 167 3.61 Variation of p(Z|^) with Incorporated in Consolidation Algorithm 172 3.62 Bed Schematization Used in Bed Formation - Consolidation Algorithms 176 4.1 Global and Local Coordinates 190 5.1 Downstream View of Recirculating Flume. Width Reducing Device is Shown on Right Side of Flume 205 5.2 Schematic Diagram of Recirculating Flume (after Dixit, 1982). 206 5.3 Kent Mini-Flow Current Meter 208 5.4 Calibration of Kent Mini-Flow Current Meter .... 208 5.5 Instrumentation Cart and Setup of Kent Mini-Flow Meter and Two Point Gages 210 5.6 Electric Point Gage and Tube of Water Surface Elevation Measuring Device 211 5.7 Setup of Water Surface Elevation Measuring Device (after Wang, 1983) 211 xi V FIGURE PAGE 5.8 (a) Apparatus I for Obtaining Sediment Core; (b) Apparatus II for Sectioning a Frozen Sediment Core (after Parchure, 1980) 213 5.9 Water Sampling Device 215 5.10 Grain Size Distribution of Kaolinite Used for the Experiments 215 5.11 Shear Stress History for Experiments in the Recirculating Flume 219 5.12 Measurement Stations in the Recirculating Flume 221 5.13 Measured and Predicted Suspended Sediment Concentrations for Test No. 1 223 5.14 Measured and Predicted Suspended Sediment Concentrations for Test No. 2 224 5.15 Measured and Predicted Suspended Sediment Concentrations for Test No. 3 225 5.16 Measured Bed Density Profiles for Experiments in the Recirculating Flume 228 5.17 Rotating Annular Flume 229 5.18 Velocity-Time Record, and the Measured and Predicted Suspended Sediment Concentrations in the Tidal Cycle Experiment 232 5.19 Finite Element Grid of Recirculating Flume; Distorted Sketch - Width: Length =4.1:1.0 234 5.20 Schematic Representation of Set up for Experiment in the 100 m Flume (after Dixit et al . , 1982) 237 5.21 Measured and Predicted Deposit Thickness Along 100 m Flume .... 239 5.22 Aerial View of Camachee Cove Marina 247 5.23 Bathymetry of Camachee Cove Marina Obtained in September, 1982 (after Srivastava, 1983). ..... 248 5.24 Finite Element Grid of Camachee Cove Marina .... 250 5.25 Predicted Sedimentation Contours for Marina Basin 251 XV t^IGURE PAGE 5.26 Plan View of 10 km Hypothetical Canal 253 5.27 Predicted Suspended Sediment Concentration-time Record for Element No. 4 in Hypothetical Canal 255 5.28 Predicted Suspended Sediment Concentration-time Record for Element No. 5 in Hypothetical Canal 256 xvi Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PREDICTION OF COHESIVE SEDIMENT MOVEMENT IN ESTUARIAL WATERS By Earl Joseph Hayter December 1983 Chairman' Dr. B.A. Christensen Co-Chairman : Dr. A.J. Mehta Major Department: Civil Engineering Fine sediment related problems in estuaries include shoaling in navigable waterways and water pollution. A two-dimensional (horizontal) fine, cohesive sediment transport model using the finite element method has been developed to predict the temporal and spatial variations of the depth-averaged suspended sediment concentrations in estuarial waters. The advection-di spersion equation with appropriate source/sink terms is solved by the Galerkin weighted residual method for the suspension concentration at each node. Contemporary laboratory and field evidence has been used to develop algorithms which describe the processes of erosion, dispersion, settling, deposition, bed formation and bed consolidation. XV i 1 The model yields stable and converging solutions. A useful feature of the model is its ability to predict the influence of salinity on the rate of fine suspended sediment movement. Verification was carried out against results from a series of erosion-deposition experiments in the laboratory using kaolinite and a natural mud as the sediments. The model was applied under prototype conditions to simulate sedimentation in a marina basin and suspended sediment transport in a hypothetical canal in which both erosion and deposition occurred. XV i i i CHAPTER I INTRODUCTION 1.1. Estuarial Cohesive Sediment Dynamics Cohesive sediments in estuaries are comprised largely of terrigenous clay-sized particles. The remainder may include fine silts, biogenic detritus, algae, organic matter, waste materials and sometimes small quantities of very fine sand. Although in water with a very low salinity (less than about 1 part per thousand) the elementary sediment particles are usually found in a dispersed or "non-salt flocculated" state, small amounts of salts are sufficient to repress the electrochemical surface repulsive forces between the elementary particles, with the result that the particles coagulate to form much larger aggregates. Each aggregate may contain thousands or even millions of elementary particles. The transport properties of aggregates are affected by the hydrodynamic conditions and by the chemical composition of the suspending fluid. Most estuaries contain abundant quantities of cohesive sediments which usually occur in the coagulated form in various degrees of aggregation. Therefore, an understanding of the transport properties of these sediments in estuaries requires a knowledge of the manner in which the aggregates are transported in these waters. Cohesive sediment transport in estuaries is a complex process involving a strong coupling between tides, baroclinic circulation and the coagulated sediment. For an extensive description of this process, 1 2 the reader is referred to Postma (1967), Parthem'ades (1971), Barnes and Green (1971), Krone (1972), Kirby and Parker (1977), and Kranck (1980). In Fig. 1.1, a schematic description is given. The case considered is one in which the estuary is stratified, and a stationary saline wedge is formed as shown. Various phases of suspended fine sediment transport are shown, assuming a ti daily-averaged situation. In the case of a partially mixed estuary, the description will be modified, but since relatively steep vertical density gradients are usually present even in this case, the sediment transport processes will generally remain qualitatively similar as depicted in Fig. 1.1. As indicated in Fig. 1.1, riverborne sediments from upstream fresh water sources arrive in the mixing zone of the estuary. The comparatively high degree of turbulence, the associated shearing rates and the increasingly saline waters will cause aggregates to form and grow in size as a result of frequent interparticle collisions and increased cohesion. The large aggregates will settle to the lower portion of the water column because of their high settling velocities. Results based on laboratory experiments show that aggregate settling velocities can be up to four orders of magnitude larger than the settling velocities of the elementary particles (Bellessort, 1973). Some of the sediment will deposit and some will be carried upstream near the bottom until periods close to slack water when the bed shear stresses decrease sufficiently to permit deposition. The sediment will start to undergo self-weight consolidation. The depth to which the new deposit scours when the currents increase after slack will depend on the bed shear stresses imposed by the flow and the shear strength of the deposit. Net deposition, i.e. shoaling, will occur when the bed shear 3 o <D o CD c: 0) x: (/) O) CO (/) OJ o o s_ D_ cn . c •t- <—l r— CO rC CD O ^ TH ID E +-> 03 >, +J S- O T3 Q. C ro ra t— -C <D O i- C OJ o -u ■r- 4- +-> ro fC w 4-) C >, CU i- c/1 ro O) :=5 +-> Q. </) O) LU -a O O) ■r— 'r— +-> H- ro -r- E +-> <D ro U -P 00 CO 4 during flood, as well as during ebb, is insufficient to resuspend all of the material deposited during preceding slack periods. Some of the sediment that is resuspended will be re-entrained throughout most of the length of the mixing zone to levels above the salt water-fresh water interface and will be transported downstream to form larger aggregates once again, and these will settle as before. At the seaward end some material may be transported out of the system, a portion or all of which could ultimately return with the net upstream bottom current. In the mixing zone of a typical estuary the sediment transport rates often are an order of magnitude greater than the rate of inflow of "new" sediment derived from upland or oceanic sources. The estuarial sedimentary regime is characterized by several periodic (or quasi - periodic) macro- time-scales, the most important of which are the tidal period (diurnal, semi-diurnal, or mixed) and one-half the lunar month (spring-neap-spring cycle). The first is of course the most important since it is the fundamental period which characterizes the basic mode of sediment transport in an estuary. The second is important from the point of view of determining net shoaling rates in many cases of engineering interest. 1.2. Sediment Related Problems in Estuaries Estuaries are often centers of population and industry, and as such are used as commerce routes to the sea, convenient dump sites for waste products as well as for man's recreational enjoyment. They also serve as the sink for sediment and pollutants transported by rivers from inland sources. As man's activity in and hence dependence upon estuaries has increased with the growth of population and commerce in 5 these areas, the necessity of proper estuan'al management becomes very ostensible. Included in estuarial management is the maintenance of navigable waterways and water pollution control, both of which are affected to varying degrees by the load of suspended and deposited sediment. These two tasks are examined next. Under low flow velocities, sometimes coupled with turbulent conditions which favor the formation of large aggregates, cohesive sediments have a tendency to redeposit in areas such as dredged cuts and navigation channels, in basins such as harbors and marinas, and behind pilings placed in water (Einstein and Krone, 1962; Ariathurai and F^ehta, 1983). In addition, as noted previously, the mixing zone between upland fresh water and sea water in estuaries is a favorable site for bottom sediment accumulation. Inasmuch as estuaries are often used as transportation routes, it is desirable to be able to accurately estimate the amount of dredging required to maintain navigable depths in these water bodies, and also to predict the effect of new estuarial development projects such as the construction of a port facility or dredging of additional navigation channels. Cohesive sediments may influence water quality by affecting aquatic life and by providing a large assimilative capacity as well as transporting mechanism for dissolved and suspended pollutants. Turbidity caused by suspended sediment particles restricts the penetration of light, and therefore reduces the depth of the photic zone. This in turn may result in a decrease in the production of phytoplankton and other algae which leads to a reduction in the amount of food available for fish. Deposited sediments can damage spawning areas for fish and eliminate invertebrate (e.g. oysters) populations. 6 The bulk of the pollution load in a water body is quite often transported sorbed to cohesive sediments rather than in the non-sorbed state (Preston etal_., 1972; Kirby and Parker, 1973). Therefore, the importance of considering the movement of cohesive sediments in predicting the fate of pollutants (e.g. pesticides, radioisotopes, and toxic elements such as lead, mercury, cadmium, nickel and arsenic) introduced in an estuary cannot be overemphasized. The properties of cohesive sediments, and in particular clays, which cause the sorption of pollutants are the large surface area to volume ratio, the net negative electrical charges on their surfaces and their cation exchange capacity. These properties are discussed in Chapter II. In an investigation of the bottom sediments from several coastal marinas in Florida, two interesting observations were made (Weckmann, 1979; Bauer, 1981). First, when comparing sediment particle size inside the basin with that obtained immediately outside in the main body of water, it was found that in the majority of the marinas investigated, the sediment inside was measurably finer than that outside. Second, a similar comparison in terms of heavy metal (e.g. Cu, Pb, Ni, Cd and Zn) content within the basin and without indicated measurably higher concentrations inside the basin. These two observations, when correlated, exemplify the role of cohesive sediments in accumulating pollutant levels in estuarial depositional environments such as marina basins. This assimilation of pollutants and storage in bottom sediments may prove to be an acceptable means of waste disposal, providing the contaminated bottom sediment is not resuspended and the pollutant desorbed. However, sometimes even a relatively small change in the chemical composition of the water may cause desorption of pollutants from the sediment particles. 7 1.3. Approach to the Problems Prediction of the fate of sorbed pollutants or the frequency and quantity of dredging required to maintain navigable depths in a channel or harbor can be accomplished by modeling the movement of cohesive sediments in the water body of concern. It becomes necessary to simulate the various transport processes, i.e. erosion, advective and dispersive transport, aggregation, deposition and consolidation, and the physical factors, e.g. movement of water and dissolved salt, that govern these processes. The movement of suspended sediment, water and salt are highly interrelated, as is evident upon examination of Fig. 1.2 which defines possible interactions between these constituents in an estuary. Physical and mathematical models or combinations (hybrid approach) of these two types are the types of models available for use in predicting cohesive sediment movement in a water body. Physical scale models have only been partially successful due to lack of an appropriate model sediment as well as due to poor model reproduction of estuarial mixing processes and internal shear stresses (Owen, 1977). Mathematical models, however, have been generally more successful in reproducing, with some degree of accuracy, the movement of cohesive sediments in estuarial waters. The modeling philosophy is delineated below. To mathematically model the motion of the three main constituents in an estuarial environment the three-dimensional forms of the conservation of momentum and mass equations for the water and the conservation of mass equations for the dissolved salt, suspended sediment and pollutant, if present, must be solved numerically. However, due to the current high cost of solving such three-dimensional, coupled, partial differential equations, only a few three-dimensional models 8 Gradual Large Motion of dissolved sal t Rapid Large Large Bulk flow tidal propagation Rapid Large Rapid aMedi um Large Rapid Velocity field. Internal water ci rculation. Bed shear Rapid Medium Rapid Large Rapid Large Medium Gradual Medi um Large Rapid Very Gradual Smal 1 Motion of suspended mud Larae Rapid Large Rapid Large Rapid Medi um Coagulation' and settling /Medium Gradual Medium Fig. 1.2. Interactions of Tidal and Estuarial Sediment Transport Processes (after Owen, 1977). 9 exist (L1u and Leendertse, 1978). The common procedure has been to spatially integrate these equations, laterally and/or vertically, in order to reduce them to their two- or one-dimensional forms. The horizontal length scales relative to the transport of cohesive sediments typically are one to three orders of magnitude greater than the vertical length scales in most estuaries. As a consequence, and because horizontal transport distances are usually of primary interest in ascertaining the magnitude of sedimentation or the fate of sorbed pollutants, it is in most cases not unreasonable to use vertically integrated transport equations for modeling purposes. However, even using the two-dimensional forms of the governing equations, some eight to ten coupled equations must be solved to completely model the depth- averaged motion of water, sediment and salt. As a result, the modeling of water and salt movement is commonly performed separately from the sediment transport modeling. For example, a two-dimensional hydrodynamic model, which solves the coupled momentum and (water and dissolved salt) continuity equations, is used to model the movement of water and salt. Then a two-dimensional cohesive sediment transport model would be used to predict the motion of sediment using the results from the hydrodynamic model. 1.4. Scope of Investigation Mathematical descriptions of the physical processes prevalent in a binary fluid-sediment system (such as the flux of sediment to and from the bed, and the dispersion, aggregation and settling of suspended aggregates in a turbulent flow field) that are incorporated in existing suspended cohesive sediment transport models, many of which are 10 described in Chapter IV, are limited as they use empirical evidence based on limited studies conducted prior to the early 1970's. Since that time, a considerable amount of experimental research has been conducted, partly at the University of Florida, on the various aspects of cohesive sediment transport mechanics. Utilization of contemporary laboratory experimental and field evidence to develop new algorithms which describe the transport processes of erosion, dispersion, settling, deposition and bed consolidation would result in a model whose predictive capability is measurably improved over that of existing model s. The intent of this investigation therefore was twofold: 1) to develop a two-dimensional depth-averaged cohesive sediment transport model using state-of-the-art information on estuarial cohesive sediment processes, and 2) to verify this model with results from erosion/deposition experiments performed in a 18 m long recirculating flume and in an annular rotating channel, and a deposition experiment performed in a 100 m long flume. The sediment process information incorporated in the model is based on a detailed analysis and interpretation of available laboratory and field data. The results of this research is presented in the following format. In Chapter II, a description of the composition, structure and physicochemical properties of cohesive sediments is given. This is followed by discussions on estuarial dynamics and sedimentation processes. Chapter III begins with a description of the governing equations for sediment transport in an estuarial environment. Then, previous investigations on erosion, dispersive transport, deposition, and the 11 structure and consolidation of cohesive sediment beds are described, followed by detailed descriptions of the sediment transport algorithms developed in this investigation. In Chapter IV, the new cohesive sediment transport model is described in detail. Included in this chapter is a review of previous models, and descriptions of the finite element solution routine and the stability and convergence characteristics of the new model. In Chapter V, the model is verified using the results from: 1) experiments performed by the investigator in a 18 m long recirculating flume at the University of Florida, and 2) a deposition experiment performed in a 100 m long flume at the U.S. Army Corps of Engineers VJaterways Experiment Station in Vicksburg, Mississippi. The model is then used to simulate the sedimentation in a coastal marina. Lastly, limitations of the modeling approach, as well as possible model applications to water quality and sedimentation management problems are di scussed. In Chapter VI, conclusions from this study and recommendations for future research are presented. In Appendix A, the two-dimensional depth-averaged form of the advection-dispersion equation is derived. In Appendix B, the element coefficient and load matrices are given. In Appendix C, a description of the computer program, including a user's manual is given. Lastly, in Appendix D, the field data collection and laboratory analysis programs required to develop the data base for the model are described. CHAPTER II BACKGROUND MATERIAL 2.1. Introductory Note Understanding the movement of cohesive sediments 1n an estuarlal environment requires knowledge pertaining to the physicochemical properties of cohesive sediments, estuarial hydrodynamics and sediment transport processes. Each of these topics is briefly discussed below. 2.2. Description and Properties of Cohesive Sediments 2.2.1. Composition As noted in Chapter I, cohesive sediments consist primarily of clay-sized material. Such material consists of clay and non-clay mineral components and organic material (Grim, 1968). Clay particles are generally less than 2 microns {\m) in size. As a result they are termed colloids, and in water possess the properties of plasticity, thixotropy and adsorption (van Olphen, 1963). Grim (1968) states that "the term clay implies a natural, earthy, fine-grained material", and that clays are "composed essentially of silica, alumina, and water, frequently with appreciable quantities of iron, alkalies, and alkaline earths." The most abundant types of clay minerals are kaolinite, montmoril lonite, illite, chlorite, vermiculite, and halloysite. Non-clay minerals consist of, among others, quartz, carbonates, feldspar, and mica (Grim, 1968). This component of clay material is generally larger than 2 i^m in size, though this is not so in all clay 12 13 mateHals. It H currently not possible to quantitatively determine, with a high degree of accuracy, the amount of non-clay minerals present in a clay mateHal. Grim (1958) points out the consequence of this 1t«ttatfon by stating that "the absence of accurate quantitative methods for determining the non-clay mineral components of clay materials frequently makes ft impossible m obtain exact data on the chemical composition of the clay minerals themselves in such materials." The organic material usually present in clay materials may sxfst as discrete particles of matter (e.g. wood), as sorbed organic molecoles on the surface of the clay ptrtfdes. or' inserted between clay layers (Gn-m. 1968). The percentage by weight of organic matter in a clay material may be determined through use of a standard analytical procedure such as the Walkley-Rlack test (Allison. 1965). Additional possible components of clay materials are water-soluble salts, and sorbed exchangeable ions and pollutants. Water-soluble salts include chlorides, sulfates, alkaline earths, carbonates of alkalies, aluminum and iron. The most common exchangeable cations and anions in clay materials are. respectively. Ca^^ Mg2^ n\ K*. mH,"^ and Cr, SO^". NO3- and P0,3- (Grim. 1968). The presence of pollutants sorbed to the surface of clay particles was discussed in Chapter 1. 2.2. g. Origin In nature clay materials are produced by hydrothermal action and weathering of rocks. Fact0rs which Influence the type of clay minerals formed by these two processes include the composition of the parent rock, m Climate, topography, the abundance and kind of vegetition, time, and pH of the ground water, the presence and kind of alkalies and 14 alkaline earths, the intensity of the hydrothermal alteration, and the permeability and porosity of the host rock. Grim (1968) describes the origin and occurrence of clay materials in detail. 2.2.3. Structure Clay minerals are primarily hydrous aluminum silicates with magnesium or iron occupying all or part of the aluminum positions in some clays, and with alkalies (e.g. sodium, potassium) or alkaline earths (e.g. calcium, magnesium) also present in others (Grim, 1968). Most clays are composed of one of two atomic structural units, or combinations of the two basic units. These are the silica tetrahedron and the aluminum hydroxide octahedral unit. The former consists of a central silicon atom surrounded by four oxygen atoms or hydroxy Is arranged in a tetrahedral configuration. The tetrahedrons are bonded together in a hexagonal network in such a way that a sheet structure of composition Si40g(0H)4 is formed. The structure of each sheet is such that the oxygens and/or hydroxy Is forming the bases of the tetrahedral units are in the same plane, and the tips of all the units point in the same direction (Grim, 1968). The octahedral aluminum hydroxide unit consists of "two sheets of closely packed oxygens or hydroxyls in which aluminum, iron, or magnesium atoms are embedded in octahedral coordination, so that they are equidistant from six oxygens or hydroxyls" (Grim, 1968). If aluminum atoms are present, only two out of every three central positions will be filled so that the structure's electrical charge will be balanced. However, if magnesium atoms are present, all central positions are occupied. With aluminum, the octahedral is known as the 15 gibbsite structure, which has the formula Alj(OH)g. The brucite structure is that formed with magnesium, and has the formula Mg3(0H)g (Grim, 1968). The different clay minerals are formed by stacking of the sheet structures to form layers, and substitution of different ions for the aluminum in the octahedral unit. Grim (1968) gives an excellent description of the structural arrangement of the most common types of clay minerals. Ions of one kind are sometimes substituted by ions of another kind, with the same or different valence. This process does not necessarily involve replacement. The tetrahedral and octahedral cation distributions develop during initial formation of the mineral, and not by later substitution (Mitchell, 1976). Substitution in all the clay materials, except for kaolinite, gives clay particles a negative electric charge which is of great significance in coagulation of clays and in absorption of pollutants. Another cause of net particle charge is the preferential sorption of peptizing ions on the surface of the particle (van Olphen, 1963). 2.2.4. Interparticle Forces For particles in the colloidal size range, surface physicochemical forces exert a distinct influence on the behavior of the particles due to the large specific area, i.e. ratio of surface area to volume. As stated previously, most clay particles fall within the colloidal range in terms of both their size (2 lim or less) and the controlling influence of surface forces on the behavior. In fact, the average surface force 6 on one clay particle is approximately 10 times greater than the gravitational force (Partheniades, 1962), 16 The relationship between clay particles and water molecules is governed by the interparticle electrochemical forces. The different configurations and groupings as well as electric charges of clay particles affect their association with water molecules (Grimshaw, 1971). Water molecules possess a permanent electrical imbalance or dipole moment which results from the molecular arrangement of the oxygen and hydrogen atoms. According to Grim (1962), the electrostatic field emanating from the surface of a clay particle orients the polar water molecules in the pores separating adjacent particles. Interparticle forces consist of both attractive and repulsive forces. The attractive forces present are the London-van der Waals, and are due to the nearly instantaneous fluctuation of the dipoles which result from the electrostatic attraction of the nucleus of one atom for the electron cloud of a neighboring atom (Grimshaw, 1971). These electrical attractive forces are weak, and are only significant when interacting atoms are very close together. However, they are strong enough to cause structural build-up as they are additive between pairs of atoms. Thus, the total attractive force between two clay particles is equal to the sum of the attraction between all the atoms comprising both particles. This additive effect results in a larger attractive force and to a smaller decrease in this force with increasing particle separation. Figure 2.1 shows qualitatively the relationship between the attractive energy V^^ of one particle for another and the particle separation distance. The attractive energy is inversely proportional to the sixth power of the separation distance for two atoms and to the second power for two spherical particles. The magnitude of decreases with increasing temperature and is dependent upon the geometry and the 17 Double -Layer Repulsion at Three Different Electrolyte Concentrations Fig. 2.1. Repulsive and Attractive Energy as a Function of Particle Separation at Three Electrolyte Concentrations (after van Olphen, 1953) . V Net Interaction Energy Particle Seoaration ■ >- — f '1 1 min Fig. 2.2. Net Interaction Energy as a Function of Particle Separation at High Electrolyte Concentration (after van Olphen, 1953). 18 size of adjacent clay particles. The attractive energy has been found to be only slightly dependent upon the salt concentration (i.e. salinity) of the medium (van Olphen, 1963). The repulsive forces of clay materials are due to the negatively charged particle forces. The repulsion potential increases in an exponential fashion with decreasing particle separation. The magnitude of these forces is dependent upon the salinity, decreasing with increasing salinity as shown in Fig. 2.1, where is the repulsive energy. This dependence of Vj, on the salinity can best be explained using the concept of the electrical double layer and the surrounding diffuse layer, van Olphen (1963) states that the double layer is composed of the net electrical charge of the elementary clay particle and an equal quantity of ionic charge of opposite sign located in the medium near the particle surface. Thus, the net electrical charge is balanced in the surrounding medium. The ions of opposite charge are called the counter-ions, i.e. cations. The counter-ion concentration increases with decreasing distance from the particle surface. This layer of counter-ions is referred to as the diffuse layer. A clay particle and the associated double layer is referred to as a clay micelle (Partheniades, 1971). When the salinity is increased, the diffuse layer is compressed toward the particle surface (van Olphen, 1963). The higher the salinity, and as well the higher the valence of the cations which compose the diffuse layer, the more this layer is compressed and the greater the repulsive force is decreased. With a high salinity, corresponding to a value approximately that of seawater (35 ppt), the attractive forces become predominant at all but extremely small particle separation distances. The interaction 19 potential, determined by summing and V^, reflects this dominance, and shows the highest attractive potential (primary minimum) at separation distances on the order of 1 nm (10~\) (Parker, 1980). At distances less than this the short range repulsive forces are predominant (van Olphen, 1963). Figure 2.2 shows this net interaction potential as a function of particle separation for high salinity. Thus, two clay particles will adhere when they reach the separation distance at which the primary minimum occurs. Cohesion or particle destabi 1 ization occurs at a maximum rate due to the presence of attractive forces even at relatively great distances. For medium and low salinities, on the order of 10-15 ppt and 1-2 ppt respectively (Parker, 1980), repulsive forces become predominant at separation distances of approximately 10 nm where a local repulsive potential maximum occurs (Figs. 2.3 and 2.4). At distances closer than this, these interaction potentials are similar to that for high salinity. As indicated by these figures, the destabi 1 ization of two or more particles would be expected to decrease for decreasing salinities as a result of net repulsive forces existing at increasingly larger distances (van Olphen, 1963). 2.2.5. Cation Exchange Capacity The cation exchange capacity (CEC) is an important property of clays by which they sorb certain cations and anions in exchange for those already present and retain them in an exchangeable state. The CEC of different clays varies from 3-15 milliequivalents per 100 grams of material (meg/100 gm) for kaolinite to 100-150 meg/100 gm for vermiculite. Higher CEC values indicate greater capacity to absorb 20 Fig. 2.3. Net Interaction Energy as a Function of Particle Separation at Intermediate Electrolyte Concentration (after van Olphen, 1953). f min Fig. 2.4. Net Interaction Energy as a Function of Particle Separation at Low Electrolyte Concentration (after van Olphen, 1963). 21 other cations. The negative surface charge caused by isomorphous substitution is neutralized by sorbed cations located on the surfaces and edges of a clay particle. These cations remain in an exchangeable position and may in turn be replaced by other cations. The following factors are the causes of cation exchange: 1) substitution within the lattice structure results in unbalanced electrical charges in the structural units of some clays, and 2) broken bonds around the edges of the tetrahedral -octahedral units give rise to unsatisfied charges. In both cases the unbalanced charges are balanced by the sorbed cations. The number of broken bonds and hence the CEC increases with decreasing particle size. The ability to replace exchangeable cations depends on the concentration of the replacing cation, the number of available exchange positions, the nature of the anions and cations in the replacing solution. Increased concentration of the replacing cation results in greater cation exchange. The release of an ion depends upon the nature of the ion itself, upon the nature of the other ions filling the remaining exchange positions, and upon the number of unfilled exchange sites. The higher the valence of a cation, the greater is its replacing power and the more difficult it is to displace when sorbed on a clay. Some of the predominantly occurring cations in sediments are sodium, potassium, calcium, aluminum, lead, copper, mercury, chromium, cadmium and zinc. 2.2.6. Coagulation Coagulation of suspended cohesive sediments depends upon interparticle collision and interparticle cohesion after collision. 22 Cohesion and collision, discussed in detail by among others Kruyt (1952), Einstein and Krone (1962), Krone (1962), Partheniades (1964), O'Melia (1972), and Hunt (1980) are reviewed here. The collision frequency, I, for suspended sediment particles of effective diameters d^- and d - is given by (Hunt, 1980): I = P(d.,d ) dN dN. (2.1) ' u 'J where P(d^-,dj) = collision function determined by the collision mechanism (discussed below), which has units of fluid volume per unit time, and dN^ = number of particles with sizes between d^- and d^-+d(d^-) per unit volume of the fluid. There are three principle mechanisms of interparticle collision in suspension, and these influence the rate at which elementary sediment particles coagulate. The first is due to Brownian motion resulting from thermal motions of molecules of the suspending ambient medium. The collision function corresponding to this mechanism is given by (Hunt, 1980): 2 kT. (d.+d.)^ Pu(d.,d.) = ^ ' ^ (2.2) ° ^ 3 ^ d.d. where k = Boltzmann constant, T|^ = absolute temperature and \^ = dynamic viscosity of the fluid. Generally, coagulation rates by this mechanism are too slow to be significant in estuaries unless the suspended sediment concentration exceeds 10 g/1 . Aggregates formed by this mechanism are weak, with a lace-like structure and are easily fractured by shearing in the flow or are crushed easily when deposited (Krone, 1962). 23 The second mechanism is that due to internal shearing produced by local velocity gradients in the fluid. Collision will occur if the paths of the particle centers in the velocity gradient are displaced by a distance which is less than the sum of their radii (referred to as the where G is the local shearing rate and R^- j = d^.+dj. Aggregates produced by this mechanism tend to be spherical, and are relatively dense and strong because only those bonds that are strong enough to resist the internal shearing due to local velocity gradients can survive. The frequency of collision is especially high in an estuarial mixing region where a large number of suspended particles are found. The third mechanism, differential sedimentation, results from the fact that particles of different sizes have different settling velocities. Thus a larger particle, due to its higher settling velocity, will collide with smaller, more slowly settling particles along its path and will have a tendency to "pick up" these particles on its way down. The collision function is expressed as collision radius, R^-j, between d^- and dj size particles). The collision function is given as P Jd.,d .) = - R^ . sh' 1' g ij (2.3) Tig p^-p^ w .)(d.+d.)^|d.^-d.^! (2.4) 11 1 J 24 where v = kinematic viscosity of the fluid, = floe density and = fluid density. This mechanism produces relatively weak aggregates and contributes to the often observed rapid clarification of estuarial waters at slack. All three mechanisms operate in an estuary, with internal shearing and differential sedimentation generally being predominant in the water column, excluding perhaps the high density near-bed layer, where Brownian motion is likely to contribute significantly as a collision mechanism. Then again, internal shearing is probably more important than differential sedimentation during times, excluding those near slack water, when collision and coherence due to differential settling would be expected to be the main mechanism controlling the rate of coagulation. Hunt (1980) compared the values of the three collision functions (Eqs. 2.2 - 2.4) for collision of a d^.=l \m size particle with varying sizes, dj, of the colliding particle under the following conditions: temperature 14°C, shearing rate G = 3 sec"-"-, and (P^'P^^fP^ ~ 0.02. The comparison is shown in Fig. 2.5 and reveals that each collision mechanism is dominant over a certain particle size range. In this example, Brownian motion is the dominant mechanism for particles less than 1 m, internal shearing is dominant for particles between 1 and 100 m, and differential sedimentation is dominant for particles greater than 100 \m. Hunt states that the same ordering of the dominant collision mechanisms with increasing dj would be achieved for collisions with other d^- sizes. Thus, the collision frequency is controlled not only by the prevailing flow conditions and local suspension concentration, but by the size of the colliding particles and/or floes as well. 25 d; (/xm) Fig. 2.5. Comparison of the Collision Functions for Brownian, Shear and Differential -Sedimentation Coagulation (after Hunt, 1980). 26 Cohesion or particle destabilization of colloidal particles is caused by the presence of net attractive electrochemical surface forces on the particles. The latter condition is promoted by Increased concentration of dissolved ions and/or increased ratio of multivalent to monovalent ions, both of which serve to depress the double layer around micelles and thus allow the attractive London-van der Waals and coulombic forces to predominate (Krone, 1963). Since sea salt is a mixture of salts, with monovalent sodium ions and divalent calcium and magnesium ions prevalent in natural electrolytes, the effect of these salts on cohesion is determined by the relative abundances of mainly these three ions (see Table 3.2), the latter being indicated by the sodium adsorption ratio (SAR). The SAR is defined as where the cation concentrations are in milliequivalents per liter (Arulanandan, 1975). The relationship between the SAR and salinity is seen in Fig. 2.6. The cation exchange capacity (CEC), salinity and SAR all serve to determine the net interparticle force and thus the potential for micelles to become cohesive. Kandiah (1974) found that the boundary between the dispersed and coagulated states for the three main clay groups, kaolinite, illite and montmorillonite, varied with the SAR, total salt concentration and pH of the solution (see Figs. 2.7a, b, and c). The dashed lines in these figures represent interpolated boundary curves for a pH range of 7.5 to 27 (/) 1/1 0) > CO a. cr ee O ^ ^3 pu o +-> NC Ull +-> • ro -— ~ •r- ■=3- 8 (/) S- +-> CTi pe •1— r-l E O 1/1 O ^ og •r- Q r— "to C O o c _J +-> rd O !- s: ID 4-3 o 13 C7> 1- oa o ro ro I — CM ro <D +-) s- 0) ro 4J 00 <4- ro -C -M •r- c • o •1- ■=d- < 4-3 ro a\ M- -p o 0) -r- c u ro o c: s_ O 13 +J ro 4-3 ro tl ro 'ro ^ >- oo <: UD CD 28 >) Q- CD • (T3 -o o m 0 -C ■1- O) fO 01 +-) •!- S_ -O O) c: c l/l r— •.- o Q nj i- I ^ O) c +-> O S- 4- •r- O fO -!-> 4- — n3 I — i/l 1/1 3 OJ 0) cn > C7) 03 S- C o r3 03 cj o o ca S- 1—1 c +-> o ■1- +-> (/) 03 s~ O) QJ I/) T- o3 Q r— S- I O) E -P O S- 4- •I- O 03 +J <4- V — 03 1 — c/i (/) 13 O) OJ cn > cn ro S- C O 3 03 o o 29 8.5, which 1s the range found in sea water at all salinities. It is evident that the boundary between the dispersd and coagulated states for these three clays are different. Kaolinite becomes cohesive at a salinity of 0.6 pnt, illite at 1.1 ppt and montmorillonite at 2.4 ppt (Ariathurai, 1974). Whitehouse et_ al_. (1960) and Edzwald et al_. (1974) reported that the cohesiveness of these micelles develops quickly at the given salt concentrations, and that little increase in coagulation occurs at higher salt concentrations, which implies that the micelles must have attained the maximum degree of cohesion. The rapid development of cohesion and the low salinities at which the main clay types become cohesive indicates that cohesion is primarily affected by salinity variations near the landward end of an estuary where salinities are less than about 3 ppt. The above cohesion mechanism is referred to as salt flocculation. There is another cohesion mechanism that operates in water between micelles in the absence of salt, and hence it is termed non-salt flocculation. However, in an estuary the conditions are conducive for destabilization to be caused by salt flocculation (i.e. depression of the diffuse double layer). Both types of destabilizing mechanisms are reported in detail by Lambe (1953). In summary, it is apparent that cohesive sediment transport in estuaries is strongly influenced by the coagulation behavior of dispersed sediment particles, which is controlled by the salinity field, velocity gradients and the concentration of suspended sediments. In particular, the salinity of the suspending fluid affects the process of coagulation in two ways: 1) elementary clay particles become cohesive when the salinity is equal to or greater than 1-3 ppt, and 2) the 30 presence of high velocity gradients in the estuarial mixing zone increases the collision frequency betv;een dispersed particles and/or aggregates. 2.3. Significance of Important Physical Factors in Estuarial Transport 2.3.1. Estuarial Dynamics The hydrodynamic regime in an estuary is governed by the interaction between fresh water flow, astronomical tides, wind-generated surface waves, surface (i.e. wind) stresses, Coriolis force, the geometry of the water body and the roughness characteristics of the sedimentary material composing the bed (Dyer, 1973). Geometry includes the shape and the bathymetry of the estuary. The geometry and bed roughness interact with the driving forces - the first five factors - to control the pattern of water motion (in particular the shear stress and turbulence structure near the bed), frictional resistance, tidal damping and the degree of tidal reflections (Ippen, 1966). The magnitude of the tidal flow relative to the fresh water inflow governs, to a large extent, the intensity of vertical mixing of the lower high density layer with the upper less dense layer. There exists in all estuaries a horizontal, i.e. longitudinal, salinity profile which decreases from the mouth to the upper reaches of the estuary. Such profiles have been measured in numerous estuaries world-wide. A few examples included here are: Cumbarjua canal, Goa, India during the dry season, i.e. October through June (Fig. 2.8); the San Francisco Bay and Sacramento-San Joaquin Delta (Fig. 2.9); the Pamlico River Estuary, North Carolina (Fig. 2.10); and the Yangtze River Estuary, China 31 Fig. 2.8. Monthly Salinity Distributions in the Cumbarjua Canal, Goa, India; Ebb; --- Flood (after Rao et al. , 1975). 32 Q. Q. QJ "H CO c c o o Q. Q- o o o o 0. CL 6jnqs>iJD|3 q5no|S ssDj5pouS- c o ■o -a o o O <1 I DlSlAOId — ailiAsuiiioo — puD|S| sddiqo — ^-^f E UJ < o z: UJ Q _j o o o LlI o Q ) N0llVaiN33N03 3aidOnH0 3 or o ■-3 C </> o +J c a; E fO &. o (0 ■a c rO >> n3 CQ O 1/1 (J • ■ !- to 1— I c o •1- JZi ■!-> O rs r— +-> o Ol s- O O! C +-I o <+- (_3 fO 0) "O LO ■I- m CTl O v-H !c s_ != E •r- QJ E CL O QJ ■r- 00 4- > I •.- +-> 5- r— CD (M OS •I" 33 DISTANCE DOWNSTREAM (km) Fig. 2.10. Salinity of the Surface Waters of the Pamlico River Estuary as a Function of the Distance from the Railroad Bridge at Washington, D.C. (after Edwald et al_. , 1974). 40 DISTANCE SEAWARD FROM JIANG ZHEN DOJG(km) Fig. 2.11. Computed Longitudinal Salinity Profile in the Yangtze River Estuary as a Function of the Distance Downstream from Jiang Zhen Dong for Two River Discharges (after Huang et , 1980). 34 (Fig. 2. 11). The existence of a longitudinal salinity gradient, or baroclinic force, implies that there could be a gravity driven upstream transport of a high density sediment suspension in the lower portion of the water column (Officer, 1981; Mehta and Hayter, 1981). Winds affect the hydrodynamic regime and mixing in an estuary by generating a surface shear stress and waves. The surface stress is capable of generating a surface current (whose magnitude will be approximately three percent of the wind speed at 9.1 m elevation (Hughes, 1956)) and a superelevation of the water surface along a land boundary located at the downwind end of the estuary (Ippen, 1966). The latter effect causes a vertical circulation cell, with landward flow at the surface and a reversed seaward flow along the bottom. This phenomenon as well increases the degree of vertical mixing. Along the banks and in shallow areas, surface gravity waves induced by the wind are capable of eroding bottom sediments. Since a tidal current of sufficient strength to transport (but not necessarily to erode the sediment by itself) suspended sediment is generally present, this material is advected and dispersed both longitudinally with the main tidal flow and sometimes laterally with secondary currents towards the deeper sections of the estuary. Wave action and in particular wave breaking substantially increase the intensity of surficial turbulence and mixing. The Coriolis force, caused by the earth's rotation, has both a radial (horizontal) and a tangential (vertical) component. The latter is generally negligible as it is linearly proportional to the vertical component of the flow velocity, which is typically an order of magnitude smaller than the horizontal velocity components. The magnitude of the 35 radial component depends upon the size of the water body. Most extra- tropical estuaries are relatively large and therefore the effect of this force on the hydrodynamic regime is measurable. Estuarial hydrodynamics are described in extensive detail in such texts as Ippen (1966), Barnes and Green (1971), Dyer (1973), Officer (1976) and Fischer et al . (1979). 2.3.2. Sediment Processes The sedimentary regime in an estuary is controlled by the hydrodynamics, the chemical composition of the fluid and the physicochemical properties of the cohesive sediment. These factors affect the processes of erosion, advection, dispersion, aggregation, settling., deposition and consolidation of the deposited bed. These processes are briefly described below, following a definition of a clay suspension. A "solution" of clay in a medium consists of a homogeneous dispersion of very small kinetic units, i.e. particles (van Olphen, 1963). When the Stokes diameter of the clay solution is less than 2 \m, the clay dispersion is usually referred to as a sol. The Stokes diameter of an arbitrarily shaped particle is determined by equating the particle's settling velocity with Stokes law for spherical particles and solving for the "equivalent spherical diameter" (Stokes diameter). When this diameter is greater than 2 ^m, the dispersion is called a suspension. However, through use, the term suspension has become synonymous with dispersion, and thus a clay suspension refers to both sol and suspension. Erosion of cohesive soils occurs whenever the shear stress induced by fluid flow over the bed is great enough to break the electrochemical 36 interparticle bonds (Partheniades, 1965; Paaswell, 1973). When this happens, erosion takes place by the removal of individual sediment particles and/or aggregates. This type of erosion is time dependent and is defined as surface erosion or resuspension. In contrast, another type of erosion occurs more or less instantaneously by the removal or entrainment of relatively large pieces of soil. This process is referred to as mass erosion or redispersion and occurs when the flow- induced shear stresses on the bed exceed the soil bulk strength along some deep-seated plane. Once eroded from the bed, cohesive sediment is transported entirely as suspended load (not as bed load) by the estuarial flow. Such transport is the result of three processes: 1) advection - the sediment is assumed to be transported at the speed of the local mean flow, 2) turbulent diffusion - driven by spatial suspended sediment concentration gradients, the material is diffused laterally across the width of the flow channel, vertically over the depth of flow and longitudinally in the direction of the transport, and 3) longitudinal dispersion - the suspended sediment is as well dispersed in the flow direction by spatial velocity gradients (Ippen, 1966). In fresh water, most clay particles are in a stabilized or dispersed state because the repulsive electrochemical surface forces between the particles prevent them from adhering to one another upon collision. In the increasingly saline conditions encountered moving seaward in estuaries, the repulsive forces are suppressed and clay particles coagulate to form floes. A systematic "build up" of floes as occurs in estuaries is defined as aggregation. An aggregate is considered to be the structural unit formed by the joining of floes. 37 The rate and degree of aggregation are two important factors which govern the transport of cohesive sediments in estuaries. Factors, besides the water chemistry and the magnitude of the surface forces, known to govern coagulation and aggregation include sediment size grading, mineralogical composition, particle density, organic content and the suspended concentration (i.e. availability) of the sedimentary material, the water temperature, height through which the floes have settled, and the turbulence intensity (represented by the shearing rate G) of the suspending flow (Owen, 1971). Given the mechanisms which influence the rate of aggregation in an estuary, the order of aggregation, which characterizes the packing arrangement, density and shear strength of aggregates, is determined by: 1) sediment type, 2) fluid composition, 3) local shear field, and 4) concentration of particles or floes available for aggregation. With regard to the second factor. Krone (1962; 1978) found that the structure of aggregates is dependent on the salinity for salinities less than about 10 ppt. Primary or 0-order floes are highly packed arrangements of elementary particles, with each floe consisting of perhaps as many as a million particles. Typical values of the void ratio (volume of pore water divided by volume of solids) have been estimated to be on the order of 1.2. This is equivalent to a porosity of 0.55, which is a more "open" structure than commonly occurs in eohesionless sediments (Krone, 1963). Continued aggregation under favorable shear gradients can result in the formation of first or higher order aggregates composed of loosely packed arrays of 0-order floes. Each succeeding order consists of aggregates of lower density and lower shear strength. Experimental 38 observations (Krone, 1963; 1978) tend to indicate the following approximate relationship between the aggregate shear strength, t^, and aggregate density, p^^, for many (although not all) sediments % = (P,-l)^ (2.6) where a and p are coefficients which must be determined experimentally for each sediment. Inasmuch as the shear field in an estuary exhibits significant spatial and temporal variations, a range of aggregates of different shear strengths and densities are formed, with the highest order determined by the prevailing shearing rate, G = du/dz, provided that: 1) the sediment and the fluid composition remain invariant, and 2) sufficient number of suspended particles are available for promoting coagulation and aggregation. The determination of -c^ and corresponding to each sediment-fluid mixture can be carried out through rheological diagrams of applied shear stress against the shearing rate. Such plots were developed by Krone (1963; 1978) with the help of a specially designed annular viscometer. An example of such a diagram is presented in Fig. 2.12, with the shear stress proportional to the dial reading on the viscometer and the shearing rate proportional to the rotation rate of the outer cylinder of the viscometer. Each order of aggregation corresponds to a given volume fraction of the aggregates (volume occupied by the aggregates divided by the total volume of the suspension) which in turn can be shown to be related to the relative differential viscosity (the viscosity of the suspension divided by the viscosity of the suspended medium). Given the viscosity of the suspending medium, the relative differential viscosity 39 a '9Niav3H nvia 40 is determined from the slope of the rheological diagram, and hence the volume fraction can be calculated. The density is then computed from the volume fraction. The intercept on the applied shear stress axis of the diagram corresponds to i;^; in Fig. 2.12 the ordinate intercept is proportional to t^^. Table 2.1 gives the orders of aggregation, cation exchange capacity (CEC), densities and shear strengths of sediment samples from five different sources. As observed in this table, the first four sediment samples are characterized by three orders of aggregation while the sediment from San Francisco Bay is characterized by six orders. The number of aggregation orders possible for a suspension of a given sediment is equal to the number of linear segments on the rheological diagram with different slopes. Thus, in Fig. 2.12, the sediment sample has two possible orders of aggregation. Krone (1963) postulated that each segment is related to a particular volume fraction and therefore to a different manner in which the same sediment can aggregate, i.e. different order of aggregation. Thus, for the suspensions of the first four sediments listed in Table 2.1, three different linear segments were obtained on the rheological diagrams, while for Bay mud, six segments, and therefore six orders of aggregation were found. This indicates that Bay mud can aggregate in three more ways than the other four sediments, and further suggests that the Bay sediment is more cohesive than the others. Also observed in this table is the very rapid decrease in the shear strengths and somewhat less rapid decrease in densities with increasing order of aggregation. These trends indicate that as the order of aggregation increases, the inter- aggregate pore volume increases and the strength of these aggregates decreases because of limited bonding area between the lower order aggregates (Krone, 1978). 41 Table 2.1 Properties of Sediment Aggregates (after Krone, 1963) Sediment Order of CEC Density Shear Strength Sample Aggregation (meq/100 gm) "^"^^ ^s^'^ ^'^^ Brunswick Harbor 0 38 1164 3.40 1 1090 0.41 2 1067 0.12 3 1056 0.062 Wilmington District 0 32 1250 2.10 1 1132 0.94 2 1093 0.25 3 1074 0.12 Gulf port Channel 0 49 1205 4.60 1 1106 0.69 2 1078 0.47 3 1065 0.18 VJhite River (salt) 0 60 1212 4.90 1 1109 0.68 2 1079 0.47 3 1065 0.19 San Francisco Bay 0 34 1269 2.20 1 1179 0.39 2 1137 0.14 3 1113 0.14 4 1098 0.082 5 1087 0.036 6 1079 0.020 42 The settling rate of coagulated sediment particles depends on, in part, the size and density of the aggregates and as such is a function of the processes of coagulation and aggregation (Owen, 1970). Therefore the factors which govern these two processes also affect the settling rate of the resulting aggregates. As noted in Chapter I, the settling velocities of aggregates can be several orders of magnitude larger than those of individual clay particles (Bellessort, 1973). Deposition of aggregates occurs relatively quickly during slack water. Deposition also occurs in slowly moving and/or decelerating flows, as was observed (see Fig. 2.13), for example, in the Savannah River Estuary during the second half of flood and ebb flows (Krone, 1972). Under such conditions only those aggregates with shear strengths of sufficient magnitude to withstand the highly disruptive shear stresses in the near bed region will actually deposit and adhere to the bed. Thus, deposition is governed by the bed shear stresses, turbulence structure above the bed, type of sediment, depth of flow, suspension concentration and the ionic constitution of the suspending fluid (Mehta and Partheniades, 1973). An important conclusion derived from extensive laboratory erosion and deposition experiments using a wide range of cohesive sediments under steady flow conditions was that under these conditions the two processes do not occur simultaneously as they do in cohesionless sediment transport (Hehta and Partheniades, 1975; 1979; Parchure, 1983). A flow-deposited bed of cohesive sediment aggregates possesses a vertical bulk density and shear (i.e. yield) strength profile which changes in time primarily due to consolidation. Secondary causes are thixotropy and associated physicochemical changes affecting 43 interparticle forces. Consolidation, caused by the gravitational force (overburden) of overlying deposited aggregates which crushes and thereby decreases the order of aggregation of underlying sediment, has been observed to occur in three phases (Migniot, 1968). During the first phase the bed consolidates quickly as the water in the bed moves upward through the interstices of the bed material. This phase has been found to last up to approximately 10 hours for cohesive sediments (Owen, 1977). During the second phase, which can last up to about 500 hours, water is expelled from the bed by percolation. The rate of consolidation during the third phase is even slower and the length of time it takes for a cohesive sediment bed to reach its final, fully consolidated state depends upon the nature of the sedimentary material comprising the bed and the chemical composition (i.e. ionic concentrations) of the bed pore water (Owen, 1977). The average values of the bed bulk density and shear strength increase and their vertical profiles change during each of these three phases. Consideration of the consolidation process is essential in modeling the erosive behavior of such beds because: 1) the susceptibility to erosion of a consolidating bed decreases with time due to the continual increase in shear strength, and 2) the vertical profile of the shear strength determines the level to which a bed will erode when subjected to excess shear, i.e. an applied bed shear stress in excess of the shear strength of the bed surface. From an Eulerian point of view, the superposition of oscillating tidal flows on the quasi-steady state transport phenomenon depicted in Fig. 1.1 results in corresponding oscillations of the suspended sediment concentration with time as shown by the Savannah River data in Fig. 44 2.13. Such a variation of the suspended load uUimately results from a combination of advective and dispersive transport, erosion and deposition. Because of the complexity of the phenomena, more than one interpretation is possible as far as any schematic representation of these phenomena is concerned. One such representation is shown in Fig. 2.14. According to this description, cohesive sediments can exist in four different physical states in a tidal estuary or sea: as a mobile suspension, a stationary suspension, a partially consolidated bed and as a settled bed. The last two are formed as a result of consolidation of a stationary suspension. Stationarity here implies little horizontal movement, although consolidation does mean that there is vertical (downward) movement. A stationary suspension, a partially consolidated bed and a settled bed may erode if the shear stress exceeds a certain critical value. Erosion of a stationary suspension is referred to as redispersion while erosion of a partially consolidated as well as a settled bed is termed resuspension. 45 (r/&)3'NOIlvyiN33N00 iN3Wia3S > ■r— r~ 03 E C > n3 CO 0) c~ 4J C •1 — o •r- _)_) 03 S- u o -M a> •r— T3 O) CO T3 CJ X! C" <D Q- Z5 00 M- O CM o J_} ro CU •r— S- O £_ > r" s- 4-> cu CL-I-J cu 1+- a -o rc n3 OJ +-> h- 00 LlJ ro I— 1 CXI 46 C C g "tn c 0) CL CO o o '(/) c -S. D cn c g "to c CD ■ a. t/7 ifi q: c O 10 . CL to -o c 'to O e- Q CD QQ "a CD CD •XD CD OD cn o •o "o to c o o K c o to c to (1) v_ o cr . ^ o Stall S- « +J c ■p ■a > o o O « ■P n3 tJ +J CO CTl M- ■— I O O I— •1- fT3 +-> fO +-> +-> CL)| QJ rO 00 +-> 0) -C S- O) Q-s: cu (J 4_) •.- <+- +-> (B o o 00 M I— t CM 05 CHAPTER III SEDIMENT TRANSPORT MECHANICS 3.1 Introductory Note The purpose of this chapter is 1) to discuss the mechanics of cohesive sediment transport and 2) to describe the algorithms developed during this investigation. The processes for which algorithms have been developed include erosion, dispersive transport, deposition, bed formation and subsequent consolidation. The chapter begins with a description of the equations which govern the depth-averaged, uncoupled movement of water and suspended cohesive sediments. 3.2 Governing Equations 3.2.1 Coordinate System A right-handed Cartesian coordinate system is used (Fig. 3.1). The positive X-axis is coincident with the longitudinal axis of the estuary and points downstream. The coordinate system origin is located at some datum below the bed level. The positive z-axis is the vertical dimension and points upward. The y-axis defines lateral distances and points from right to left. 3.2.2 Equations of Motion The equations which govern the two-dimensional, depth-averaged unsteady turbulent movement of an incompressible viscous fluid are statements which express two of the basic principles of Newtonian 47 48 49 physics, that of conservation of mass (continuity equation) and the conservation of momentum (equations of motion). These equations are solved numerically in order to describe the velocity field in the estuary or other water body of interest. Alternately, the velocity field may be measured in a physical scale model of the estuary. However, in this study of the "uncoupled" movement of cohesive sediments, these governing equations are not solved as it is assumed that the velocity field is known beforehand. The continuity equation and the two equations of motion are included and discussed here for the sake of completeness. 3.2.2.1. Continuity The conservation of mass, as expressed by the continuity equation, states that the mass of an incompressible fluid entering a control volume per unit time is equal to the sum of the fluid mass leaving the control volume plus the change in volume of the control volume. The depth-averaged continuity equation for an incompressible fluid is ad a a — +— (u.d)+ (v.d) = 0 (3.1) at 9x ay where d = depth of flow and u, v = time and depth-averaged water velocity components in the x- and and y - directions, respectively. 3.2.2.2. Conservation of Momemtum The conservation of momentum (which is Newton's second law of motion) for an incompressible fluid states that the product of the fluid mass and acceleration is equal to the sum of the body (gravitational) 50 forces and the normal (pressure) and tangent (friction) surface forces which act on the boundaries of the water body. The two-dimensional, depth-averaged equations of motion for an incompressible viscous fluid, which can be derived from the Navier-Stokes equations, are given by Qu 5li Qu 1 1 5 a — + u — + V — = + — [ — -I — ^ 1 + 2wvsin<j) - at dx ay dx 5x ay (uSv^)/2+ _L_a cos(e) C^d p d w av av av i ap i a 5 _+u— +v— = -+ — [— T + — T 1 - 2oJusin4> + at ax ay p^ dy p^ dx ay (3.2) 2 2I/0 Pa'^a 2 — (u +v^)^2+^V%sin(e) (3.3) 8d p d w az in which: p = pressure force P^^ = fluid density Pg = air density ■^jj = horizontal turbulent shear stresses w = angular velocity of the earth <t> = local latitude 51 g = acceleration due to gravity f = Darcy-Vieisbach friction factor Vg = wind speed at a reference elevation above the water surface 0 = angle between the wind direction and the positive X-axis = wind drag coefficient Equation 3.4 is the hydrostatic equation which results when the vertical component of the flow velocity and acceleration are small relative to the horizontal flow velocity and acceleration. The third term on the right hand side of Eqs. 3.2 and 3.3 is the Coriolis acceleration in the Northern hemisphere in the x- and y- directions, respectively. The fourth and fifth terms on the right hand side of Eqs. 3.2 and 3.3 represent the effects of bottom shear stresses and surface wind shear stresses in the x- and y- directions, respectively. The three terms on the left hand side of Eqs. 3.2 and 3.3 represent the substantive fluid acceleration in the x- and y- directions, respectively. 3.2.3. Advection - Dispersion Equation The principle of conservation of mass with appropriate source and sink terms describes the advective and dispersive transport of suspended sediment in a turbulent flow field. In this law, expressed by the advection-dispersion equation, the time-rate of change of mass of sediment in a stationary control volume is equated to the spatial rate of change of mass due to advection by an external flow field plus the spatial rate of change of mass due to diffusion and dispersion 52 processes. Both the three-dimensional form and the two-dimensional, depth-averaged forms of the advection-dispersion equation are derived in Appendix A. The latter is given here: 9 5 a d ac ac — (dC) + u— (dC) +v — (dC) = — {dD — + dD — } + at ax ay ax ^^ax ^^ay a ac ac — {dD — + dD — } + (3.5) ay y'^ax yyay T where: C = mass of sediment per unit volume of water and sediment mixture D^-j = effective sediment dispersion tensor Sj = source/sink term. Implicit in Eq. 3.5 is the assumption that the suspended material is advected in the x- and y- directions at the respective water velocity components. This assumption is reasonable for sediment that is not transported as bed load since rolling and saltation of the sediment, which occurs during bed load transport, can cause a significant difference between the water and sediment velocities. Sayre (1968) verified that this assumption is approximately true for sediment particles less than about 100 m in diameter. The source/sink term in this equation can be expressed as dC dC S = ( — I + — I )d + S (3.6) 53 dC, where — L is the rate of sediment addition (i.e. source) due to erosion dC from the bed, and —l^j is the rate of sediment removal (i.e., sink) due . . dC, dC to deposition of sediment. Expressions for — L and — L are qiven dt ^ dt^ respectively in Sections 3.4.3 and 3.6.3. S[_ accounts for the removal (sink) of a certain mass of sediment, for example, by dredging in one area (e.g. navagational channel) of a water body, and the dumping (source) of the sediment as dredge spoil in another location in the same body of water. In the following section, the schematization for sediment beds is described. This description is preceded by a general discusson on the nature (i.e. structure) of these beds as revealed in several laboratory investigations. 3.3 Sediment Bed 3.3.1 Bed Structure Surficial layers of estuarial beds, typically composed of flow- deposited cohesive sediments, occur in three different states: stationary suspensions, partially consolidated (or consolidating) beds and settled (or fully consolidated) beds. Stationary suspensions are defined by Parker and Lee (1979) as assemblages of high concentrations of sediment particles that are supported jointly by the water and the developing skeletal soil framework, and which have no horizontal movement. These suspensions, which may be regarded as extremely under consolidated soil, develop whenever the settling rate of concentrated mobile suspensions exceeds the rate of self-weight consolidation (Parker and Kirby, 1982). They tend to have a high water content (therefore low bulk density) and a very low, but measurable, shear strength, v, ^^^^ 54 must be at least as high as the bed shear, Xj^, which existed during the deposition period (Mehta et al_., 1982a). Thus, they exhibit a definite non-Newtonian rheology. Kirby and Parker (1977) found that stationary suspensions have a surface bulk density of approximately 1050 kg/m^ and a layered structure. Krone (1963) found that, in addition to the bed shear, the structure (or framework) of these suspensions depends on the aggregate order in the following manner: if the aggregates deposit without being broken up by the bed shear, the surficial layers of these suspensions will be composed of an aggregate network whose order is one higher than that of the individual settling aggregates; therefore, these layers will have lower bulk densities and shear strengths than those of the aggregates which form them. Whether or not entrainment of these suspensions, also referred to as redispersion (Parker and Kirby, 1977) and mass erosion (Paaswell, 1973), occurs during periods of erosion depends upon the mechanical shear strength (i.e. stability) of this aggregate network. That portion which remains on the bed undergoes: 1) self-weight consolidation, due to overburden pressure resulting from the weight of the overlying sediment which crushes the aggregate network below, and 2) thixotropic effects, defined as the slow rearrangement of deposited aggregates attributed to internal energy and unbalanced internal stresses (r^itchell, 1961), both of which reduce the order of aggregation of the sub-surface bed layers. This implies that the bed becomes stratified with respect to bulk density and shear strength, with both properties typically increasing montonically with depth, at least under laboratory conditions (Mehta et ail_., 1982a). Stationary suspensions generally have a lifespan that varies from a few hours to a few days. Differential 55 settling caused by sorting processes is another cause of stratified bed formation. Continued consolidation eventually results in the formation of settled mud, defined by Parker and Lee (1979) as "assemblages of particles predominantly supported by the effective contact stresses between particles as well as any excess pore water pressure." This portion of the bed has a lower water content, a lower order of aggregation, and a higher shear strength and therefore is better able to resist high bed shear stresses. The settled mud in the Severn Estuary and Inner Bristol Channel, United Kingdom, has a bulk density range from 1,300 to 1,700 kg/m^ (Kirby and Parker, 1983). In this study the primary characteristic used to distinguish between a stationary suspension and a partially consolidated or settled bed is the mode of failure that occurs when the surface of the suspension or bed is subjected to an excess shear stress (i.e. i^i^ > t;^.). Erosion by particle by particle or aggregate by aggregate removal is not a correct representation in areas where stationary suspensions exist (Kirby and Parker, 1983). As stated previously, stationary suspensions undergo redispersion while partially consolidated and/or settled beds undergo resuspension (Parker and Kirby, 1977) or surface erosion (Paaswell, 1973). Both erosion processes are discussed in Section 3.4.1. The nature of the density and shear strength profiles typically found in flow-deposited cohesive sediment beds has been revealed in laboratory tests by, among others, Richards _et_^. (1974), Owen (1975), Thorn and Parsons (1980), Parchure (1980), Bain (1981) and Dixit (1982). A review of this subject is given here. 56 Figure 3.2 shows the dimensionless density profile measured by Owen (1970) after 4.2 and 8.3 hours of consolidation for mud obtained from the Thames near Dagenham, England. The indeterminate effect of salinity on the density profile for two different beds after 67 hours of consolidation for mud obtained at the entrance to the Royal Edwards Docks, Avonmouth, Bristol, England is seen in Fig. 3.3. In fact, despite the difference in mean density the same dimensionless profile is drawn through the data points in both Figures 3.3(a) and 3.3(b). The only distinguishable difference between these two sets of density profiles is the slightly lower relative surface densities in the bed formed by settling of the lower initial suspension concentration of 7.72 g/1. The Avonmouth mud is composed predominantly of illite, which is a relatively inert clay mineral, and was found to have a CEC value of 17 meq/100 gm. Therefore, it is not surprising that salinity had very little effect on the bed density. Figure 3.4 shows the measured dimensionless density profiles for four different bed thicknesses after 67 hours of consolidation for the Avonmouth mud (Owen, 1970). The same average profile drawn in Figure 3.3 was drawn on this figure as well. A remarkable fit obtained between the data and this common dimensionless profile is evident. Figure 3.5 shows the dimensionless density profiles obtained by Thorn and Parsons (1980) after two days of consolidation for muds from the Forth Estuary at Grangemouth, Scotland, the Brisbane River at the Port of Brisbane, Australia, and the dredged channel to the Port of Belawan, Sumatra, Indonesia. The percentage of clay minerals, percentage of non-clay minerals and the cation exchange capacity for the Grangemough mud, Brisbane mud and Belawan mud were 51%, 50% and 75-80% 57 > CO -z. Q < UJ >- CO 2: u Q H/ (^z-H) 0) to T3 O C o o +J c s- 0) o 3 o 4- 3 o to o s_ Q. >) 4- 5 • l/l o O) cn Q <— I T3 " CQ OJ 3 XJ o O) 5- s- (/) +-> n3 4- O) ro cn •r- LL. 58 59 o z Q O uj — > u cr O 0 O CO I/) o CD CO r— S- •I- ra M- D- O S- "O Q. C >> +-> c •1- S- co O dJ 1— Q co O) 1/1 +-> O) <4- I — ft5 C O •I- -a Wl O) • C CQ -— ~ QJ C E "O CO •r- 3 CTi Q >-l CO H/{^z -H) o CO ■i! o£o.£ Q U3 o i£) O CD CM eg lO in 1^ o CO d 03 d o d H/(^z-H) 10. a. O o M- -a O) t/1 CO +-> o c r-. s_ ^ (D o M- C >>T- QJ +-> Q 3 ■r- O CO Q ■o CQ M- -a +J > CO CO O) "Si- OS 60 (clay minerals), 39%, 50% and 20% (non-clays) and 20 meg/100 gm, 35 meg/100 gm and 25 meg/100 gm (CEC), respectively. Thus, the Grangemouth mud is the least cohesive and the Brisbane mud is the most cohesive. This is not unexpected as the Brisbane mud has the highest percentage of montmorillonite (approximately 60% of the clay mineral fraction), which is a very active (cohesive) clay mineral. Figure 3.6 shows the dimensionless density profiles (normalized with respect to the initial suspension concentration) measured by Parchure (1980) for commercial grade kaolinite after 24, 40 and 135 hours of consolidation. This clay had a CEC of approximately 9 meq/100 gm. Figure 3.7 shows the dimensionless density profile for mud from the Mersey Estuary, England after 48 hours of consolidation (Bain, 1981). This mud was composed of 76% clay minerals and 24% silica. Figure 3.8 shows dimensionless density profiles of two different beds of the same natural mud after 48 hours of consolidation (Thorn, 1981). The solid line profile was measured using a nuclear transmissometer, while the discrete point profile was determined using a layer-by-layer sampling technique. Figure 3.9 shows the dimensionless density profiles found by Dixit (1982) for flow-deposited beds of commercial grade kaolinite after 2, 5, 11 and 24 hours of consolidation while Figure 3.10 shows these profiles after 48, 72, 96, 144 and 240 hours of consolidation. It is evident from all these density profiles that a static or dynamic deposited cohesive sediment bed has 1) a characteristic elongated^ slightly reversed S shape density profile that generally increases monotonically with depth and that is independent of the bed 61 Fig. 3.6. Variation of Bed Density v/ith Depth for Three Different Conditions of Flow Deposited Beds (after Parchure, 1980). 62 H/{^Z-H) 4- O M- O S- r-H Cl. CO cn in c s- o Q -a h- s_ CQ OJ +-> to 4- <D c: ■o o CQ to -o OJ E •r— Q n} CO ro Fi i- Ol +-> 4- n3 o 0) S- 3 o o S- CM >1 II -M U CO T3 c <D Q +-> Ul •r— i/l 3^ O) r— C =3 CO o •r— 1—1 (/) OJ " gj CO c S- -r- •p— OJ fO Q CQ • CO cy> •r- Ll_ H/r2-H) 63 1 1 1 If) CO CVl CO o "nT 1^ CT> _ ^ — OvJ 0 <j a no Q CO CP □ °<1 □ 000 □ <i t> o 0 1 1 1 1 CO d d o d cvi Q <M '3- s- o 4- CM (/) CO o II i- Q- Ar CO -P £- 13 O c OJ s a o on -a OJ (U CO -a c OJ (0 o -p •1— (/) c c o •> •r— a o r-f ov :3 (\j — CM o b o <: Q o t> o O O a o ■3r d d o CM o □ cm' Q CM od o s_ 1—1 o 1— 1 M- (/) LD 0) rt •p— CM ^- o II s_ Q. UL. >> 1— -M (/) C cu Q CO c/l -a to OJ to CU CQ CD O o +-) •r- •T— to CM o c cn H/(^z-H 54 thickness and 2) a very Tow, generally indeterminate surface density. The variation of the density profile with consolidation time is examined in Chapter III, Section 3.7. Parchure (1980) made the following observation with regard to the shear strength profiles, '^(•i^) , in flow-deposited cohesive sediment beds: '^^(z) increases rapidly with distance below the water-sediment interface for z < i^^, at which i^ = t^j^, where t^i^ (defined in Section 3.4.1) is a characteristic value of -r^. For z > z^^, "^^iz) continues to increase but at a greatly decreased rate (Figure 3.11). The influence of such a '^(-iz) profile on the erosion rate is discussed in the following section. Figure 3.12 shows a '^(^iz) profile found by Dixit (1982). From the tests conducted by Dixit, the following two observations may be made: 1) such a "^(.iz) profile was not found in five out of nine experiments, and 2) the sediment beds used by Dixit were up to six times thicker than those used by Parchure. Therefore, the t^(.(z) profiles measured by Dixit are naturally more representative of estuarine beds, and as such it is believed that until further studies are conducted, no definitive statement regarding the precise nature of "^(.{z) profiles in cohesive sediment beds can be made. The possibility of a correlation between the bed density and shear strength of cohesive sediment beds is examined in Section 3.7. 3.3.2. Effect of Salinity on Bed Structure For most cohesive soils the interparticle and interfloc contact is considered to be the only significant region between particles where normal stresses and shear stresses can be transmitted (Mitchell et al.. 65 E £ u o < Li. tr UJ Expt. 17 =0.05 H/m T^,= 24 hrs Tch = 0.2l N/m^ hJ < I I- Z u o UJ cn $ O -J LU OQ X o. Q Expt. 18 Tb =0.015 N/m^ V 40 hrs T. =0.28 N/m' cn Expt. 19 Tt, = 0 N/m' T^= 135 hrs T^,^=0.34N/m Fig. 3.11. BED SHEAR STRENGTH (N/m^) Bed Shear Strength Profiles for Kaolinite Beds (after Parchure, 1980). Fig, 3.12. Bed Shear Strength Profile for a Kaolinite Bed (after Dixit, 1982). 67 1969). In particular, it seems very likely that the primary role of the double-layer interaction and other physicochemi cal forces is to control the structure of the soil and to alter the transmitted stresses from what they would be due to the applied flow-induced shear and overburden normal stresses alone. Two factors that effect the structure of a cohesive soil, swelling and permeability, and the effect salinity has on these factors are discussed next. The degree of swelling which occurs when a soil is immersed in a fluid is Influenced by factors such as the amount of clay, shape and size of the particles, the salinity and the sodium adsorption ratio (SAR) of the eroding and pore fluid, and the presence of an imposed load on the swelling areas (Grimshaw, 1971). Sargunam et aK (1973) state that decreases in the salinity of the eroding fluid or increases in the SAR cause the surface clay particles to swell more. This swelling causes a weakening of the interparticle attractive forces and thus increases the susceptibility of the soil to erosion. Increasing the salinity of the eroding fluid causes a greater compression of the diffuse layer, thereby reducing the repulsive forces of soil particles,. This reduction serves to limit the amount of swelling. Sargunam et_al_. (1973) found that when the salinity of the eroding fluid is greater than that of the pore fluid, the yield strength of the soil is greater and therefore the erosion potential is decreased. In this case the osmotic pressure gradient across the fluid-bed interface may result in deswelling, or consolidation of the bed sediment particles, which would cause an increase in the interparticle bonding forces and therefore lessen the susceptibility to erosion. 68 It 13 believed that while this phenomenon of swelling influences to some degree the structure and hence the erosion potential of a cohesive bed, it is not nearly as significant as the upward flux of pore water due to gravitational forces in a consolidating mud. Quirk and Schofield (1955) found that the degree of permeability of clay soils depends upon the nature and the concentration of the cations present in both the eroding and pore fluids. In particular they found that permeability increased with an increase in the salinity of the eroding fluid. Swelling, stabilization (i.e. decoagulation) and consolidation are generally considered to be the main reasons for changes in the permeability. The former can cause either partial or total blockage of soil pores which would result in a decrease in permeability. Stability essentially occurs during swelling when the clay particles have separated to the extent that the interparticle repulsive forces are dominant over the attractive forces. Since an increase in the salinity of the eroding fluid serves to limit the amount of swelling which occurs and thus restricts the amount of stabilization as well, an increase in salinity would result in increased permeability. The converse was found to occur as well since, as stated previously, a decrease in the salinity of the eroding fluid causes an increase in the degree of swelling and stabilization. As mentioned previously, the effect the salinity of the pore fluid has on the bed density could be expected to be a direct function of the cation exchange capacity of the sediment. Salinity was seen (in Figure 3.3) to have very little effect on the density profile for the relatively inert Avonmouth mud. Figure 3.13 shows the indeterminate effect of salinity on the bed density profile for mud from Lake Francis, 69 70 Nebraska, of which 50% was finer than 2 m (clay-sized particles), with montmorinonite, illite, kaolinite and quartz being the predominant minerals, and with a CEC of 100 meq/ion gm. This high CEC value indicates a higher percentage of montmorillonite than the other two clay minerals. Evidently, salinity and the CEC value had very little effect on the bed density profile. Using the method described, by Mehta et_al_. (1982a), the bed shear strength (or the critical shear stress for erosion), t^^, of Lake Francis mud as a function of depth below the initial bed surface, z^, was determined as a function of salinity for salinities from 0 to 10 ppt (Figure 3.14). Two trends are observed in this graph. First, increases with depth in the upper part of the bed for all salinities (no definite data could be obtained for the lower part of the bed, i.e. for 0-0 0.2 0.4 0.6 0.8 BED SHEAR STRENGTH, (N/m^) Fig, 3.14. Bed Shear Strength Profiles as Functions of Salinity. 71 > 0.5 cm, inasmuch as this portion of the bed did not erode during these experiments). Second, increases with increasing salinity from 0 to 2 ppt; thereafter, for salinities up to 10 ppt, no measurable increase in '^^ occurred. 3.3.3. Bed Schematization To facilitate the modeling of changes in the bed surface elevation due to erosion, deposition and consolidation processes, the bed is treated in the following manner: 1) it is discretized into a number of layers and 2) the bed properties, e.g. thickness, are assumed to be spatially (in the x-y plane) invariant within each element, but not so from element to element, in order to account for inter-element spatial variances in shoaling and/or scouring patterns. These two factors are expounded upon below. The bed in each element is considered to be composed of two sections: 1) the original, settled (consolidated) bed that is present at the start of modeling and 2) new deposits located on top of the original bed, that result from deposition during the modeling. Each of these two sections is divided into a number of layers in order to specify the actual shear strength and bulk density profiles in the model. The new deposit bed section is subdivided into two sub-sections, the top referred to as unconsolidated new deposit (UNO) layers and the bottom as partially consolidated new deposit (CND) layers (Figure 3.15). The former sub-section, i.e. the one corresponding to a stationary suspension, is considered to undergo redispersion while the latter, i.e. the partially consolidated bed, undergoes resuspension when subjected to an excess shear stress. The settled bed as well undergoes 72 © Unconsolidated New Deposit (UND) Partially (2) Consolidated ^ New Deposit (PCND) ^ ® ® ® Tu Bed Surface TuNDJ^TLAYMd)) ^ ' T UND, 2 NLAYTM = 3 T, UND, 3 TcND.i (TLAY(I)) New Deposits CND,2 CND, 3 NLAYT=4 T, CND, 4 To, I (THICKO(I)) T 0,2 Settled Bed T 0,3 NLAY0=4 To,4 Fig. 3.15. Bed Schematization used in Bed Fonnation Algorithm. Fig. 3 BED SHEAR STRENGTH , ( N/m^) 0.2 16. Hypothetical Shear Strength Profile Illustrating Determination of Bed Layers Thicknesses. 73 resuspension. The number of layers indicated in Fig. 3.15 for each of the three bed sections are not fixed, as each section can be assigned any given number of layers. Stationary suspensions are represented in the depth-averaged model as being the top section of the layered bed model, even though they are not a true bed or soil, in order to account for the subsequent redispersion and/or consolidation of these suspensions. However, the time varying thickness of the bed in each element is equal to the sum of only the NLAYT CNO layers and the NLAYO settled bed layers. The following bed-related parameters are required for bed schematization in the model : 1) The bed shear strength profile in the UND. This can be ascertained from laboratory erosion tests using samples of the sediment from the water body being modeled (see Appendix D, Section D.2). 2) The number of UND layers (NLAYTM) and the thickness of each layer (TLAYM(I), 1=1, MLAYTM). These parameters must be determined using the shear strength profile. For example. Figure 3.16 shows a hypothetical t^(z) profile and illustrates that this bed section must be divided such that the variation of within each layer is approximately linear. The values at the NLAYTM+1 nodes need to be read into the model. 3) The dry sediment density values at the NLAYTM+1 nodes as well need to be determined. The p(z) profile may be determined using a laboratory freeze-dryi ng method (appropriately modified for field samples, where necessary) described by Parchure (1980), the pumping method described by Thorn and Parsons (1977), a gamma-ray nuclear transmission densitometer (Whitmarsh, 1971) or a nondestructive X- ray technique (Been and Sills, 1981). 74 4) The same parameters for the CND layers and the settled bed layers must be determined, with the settled bed parameters determined for each element (where an original bed exists). The parameters read in for the UND and CND layers are assumed to be constants for all elements. 5) A stationary suspension and/or partially consolidated bed present on top of the settled bed at the start of the modeling is simulated by reading in the dry sediment mass per unit bed area of such new deposits obtained from every element where such exist. The bed level at which the dry sediment density is approximately 480 kg/m^ is usually taken to be the top of the settled bed. Thus, the sediment located above this level is considered to be new deposits. Another method which may be used to differentiate between new deposits and the settled bed is described in Appendix D, Section D.2. Included in Appendix D is a brief description of how these various bed properties can be determined through the use of a field data collection program and a laboratory testing program. Other parameters characterizing the rate of resuspension that each layer undergoes when subjected to an excess shear must be evaluated as well; these are discussed in Section 3.4.3. The following procedure was developed for forming the new deposit bed layer(s) which result from new deposits initially present on top of the settled bed and/or deposited during the modeling. The dry sediment mass per unit bed area per element, Mg, read in initially if a new deposit exists in any given element, or deposited during modeling (as determined by the deposition algorithm) is used in conjunction with the UND and CND properties to solve iteratively for the 75 thickness of bed formed by Mq for each element where > 0. This thickness, at, depends on the dry sediment density profile, p(z), for the UND and CND layers. The thickness AT is determined using the following relationship: J* P(2)zdz AT = ~ (3.7) where * = AT ± 0.02AT. If AT is greater than TLAYM(l) (see Figure 3.15) then more than one layer of UND is added. The assumed linear variation of p within each layer is used in the above equation. When or if the UND layers are filled, the same procedure is used to fill up the CND layers below the UND layers. The bottom CND layer can never fill up; therefore, continuing deposition is accounted for by increasing the thickness of this layer, while the thicknesses of the overlying UND and CND layers remain the same. This particular filling sequence was used in order to account for the consolidation of the sediment bed due to overburden pressure during the bed formation phase by virtue of the increasing and p values with bed depth. In the next section, a discussion on the erosive behavior of fine sediment beds is given, followed by a description of the erosion algorithm. 3.4 Erosion 3.4.1. Previous Investigations Of interest in this study are the erosion (resuspension) characteristics of saturated, flow-deposited cohesive sediment beds. A number of laboratory investigations were carried out in the sixties and 76 early seventies in order to determine the rate of resuspension, e, defined as the mass of sediment eroded per unit bed surface area per unit time, as a function of a bed shear stress in steady, turbulent flows. An important conclusion from these tests was that the usual soil indices such as liquid or plastic limit do not adequately describe the erosive behavior of these soils (Mehta, 1981). For example, Partheniades (1962) concluded that the bed shear strength as measured by standard tests, e.g. the direct-shear test (Terzaghi and Peck, 1960), has no direct relationship with the soil's resistance to erosion, which is essentially governed by the strength of the interparticle and inter- aggregate bonds between the deposited sediment material. Shown in Table 3.1 are various physicochemical factors known to govern the erosive properties of these beds. These factors must be specified to properly characterize the erosive behavior. The hydrodynamic factors define the erosive forces while the bed and fluid physicochemical properties determine the resistivity of the bed to erosion. The erosive forces, characterized by the flow-induced instantaneous bed shear stress, are determined by the flow characteristics and the surface roughness of the fluid-bed interface. The sediment composition, pore and eroding fluid compositions and the structure of the flow- deposited bed at the onset of erosion must be determined in order to properly define the erosion resistance of the bed. Sediment composition is specified by the grain size distribution of the bed material (i.e. weight fraction of clays, silts), the type of clay minerals present, and the amount and type of organic matter. The CEC can be used to characterize clay composition using the apparent dielectric constant measured at selected frequencies. Each clay tested appeared to have a 77 Table 3.1 Principle Factors Controlling Erosion of Saturated Cohesive Sediment Beds HYDRODYNAMIC FACTORS (Erosive Force) BED SHEAR STRESS 'Flow Characteristics •Bed - Fluid Interface BED AND FLUID PROPERTIES (Resistive Force) SEDIMENT COMPOSITION PORE FLUID COMPOSITION ERODING FLUID COMPOSITION ■Clay Mineral Type] Ion Exchange Capacity •Clay Percentage by Weight ■Organic Matter Mono-and Divalent Cations Concentrations ] Conductivity Relative Abundance of \ car l(Na"*',Ca"^.Mg^) Mono-and Divalent Cations/ ^'^'^ v. , . y Temperature pH Salinity (NaCI,CaCl2,MgCl2) ■Temperature •pH Cementing Agents (Iron Oxide, etc) BED STRUCTURE /Placed Bed ■Stress History I D3pQ3i^3d 78 characteristic value of a "dielectric dispersion" parameter determined from these measurements. The dielectric dispersion has been defined as the amount of decrease in the apparent dielectric constant with frequency (Alizadeh, 1974). The composition of the pore and eroding fluid are specified by the temperature, pH, total amount of salts and the type and abundance of ions present, principally CI", Na+, Ca^^, and Mg2+. Cementing agents such as iron oxide can significantly increase the resistance of a sediment bed to erosion. Measurement of the electrical conductivity is used to determine the total salt concentration. The physicochemical aspects pertaining to the aforementioned factors have been summarized by Sargunam et _al_. (1973), Kandiah (1974), Arulanandan et_ al_. (1975) and Ariathurai and Arulanandan (1978). The effect of the bed structure, specifically the vertical dry sediment density and shear strength profiles, on the rate of erosion is discussed by Lambermont and Lebon (1978) and f^ehta et al_. (1982a). Several different types of relationships between the rate of erosion, £, and the time-mean value of the flow-induced bed shear stress, Xj^, have been reported for non-stratified beds. These include statistical-mechanical models (Partheniades, 1965; Christensen, 1965), a rate process model (Paaswell, 1973; Kelly and Gularte, 1981) and empirical relationships (Ariathurai and Arulanandan, 1978). These relationships typically have the following general form (Mehta, 1981): £ = ^i\,'nj^,'n2,...,'n.) (3.8) where ili,^!^, . . . ,ti^. are parameters that specify the bed resistivity. 79 The resuspension rate, £, is related to the time-rate of change of the suspension concentration, dC/dt, and to the time-rate of change of the depth of erosion, Zj^, with respect to the original bed surface elevation^, by the following expressions: dC e = d— (3.9) dt dC 1 dz. — = -p(z.)— (3.10) dt d ° dt Figure 3.17 shows the general nature of laboratory determined relationships found by, among others, Partheniades (1962) and Christensen and Das (1973) for placed beds. Placed beds are sediment beds that are formed artificially. Such beds include those that are remolded and/or compacted after placement in a test apparatus (Mehta and Partheniades, 1979). In these beds, the shear strength and density profiles show much less significant stratification over the depth of the bed than in flow-deposited beds (Mehta et_ al_. , 1982a). Shown in Figs. 3.18 and 3.19 are examples of this relationship, which may be expressed as ^ = ^^^V^ch^ ^ ^ch ^'-''^ 80 Bed shear stress Fig. 3.17. Laboratory Determined Relationship Between Erosion Rate, e, and Bed Shear Stress, (after Mehta, 1981). ^ I 1 1 I 1 i 1 1 1 1 1 Fig. 3.18. Example of Relationship Betv/een e and x, (after Mehta, 1981). 81 Fig. 3.19. e - T^t) ^^^^ °^ Partheniades (1962), Series I and II (after Mehta, 1981) . 82 where 'A = slope, and the subscript ch refers to a characteristic value (flehta, 1981; Hunt, 1981). For x^ < x^^, M = and for t > t^^, M = Mg. Thus, Eq. 3.11 has the general form z = e {x^, x^^, z^^, M^, in Eq. 3.8. The parameter x^^^ determined by extrapolation of the M2 line to the £ = 0 axis, has been interpreted to be the critical shear stress for erosion (Partheniades, 1962; Gularte, 1978). The characteristic shear strength, t^^^ is defined to be the value of the bed shear stress at which the M^^ and lines intersect. Values of s^.^, '''ch' '^1 ^^2 largely determined by the physicochemical factors given in Table 3.1. Ariathurai and Arulanandan (1978) found the same general relationship for remolded beds as given in Eq. 3.11, but with = M^. Thus, Eq. 3.11 becomes b cr £ = M (- cr (3.12) where M = M't^^^. Figure 3.20 gives an example of this relationship, with ot^ = 1/m'. Figure 3.21 shows the measured variation of C with time typically found by several investigators (Partheniades, 1962; Mehta and Partheniades, 1979; Mehta et_ al_. , 1982a) in laboratory resuspension tests with flow-deposited (stratified) beds under a constant applied ■^5. As observed, dC/dt is high initially, decreases monotonically with time and appears to approach zero. The value of x^ at the depth of erosion at which dC/dt, and therefore e becomes essentially zero has 83 2.0 't5 Ariathurai and Arulanandan ( 1978) 30% illite 20 Fig. 3.20. Dimension! ess e - t Relationship Based on Results of Ariathurai and Arulanandan (1978) (after Mehta, 1981) -1x1 6 8 10 12 14 16 18 20 22 24 TIME (Hours) Fig. 3.21. Relative Suspended Sediment Concentration Versus Time for a Stratified Bed (after Mehta and Partheniades , 1979). 84 been interpreted to be equal to (Mehta et aT_. , 1982a). This interpretation is based on the hypothesis that erosion continues as long as \ > T^, i.e. the excess shear stress x^-t^ > o. Erosion is arrested at the bed level at which x^-x^ = o. This interpretation, coupled with measurement of p(z^) and the variation of C with t can result in an empirical relationship for the rate of erosion of stratified beds. Resuspension experiments with deposited (stratified) beds were performed by Parchure (1980) in a rotating annular flume and by Dixit (1982) in a recirculating straight flume. Both flumes are described in Chapter V, Section 5.2. The objective of these experiments was to determine the effect of varying bed shear strength with depth below the initial bed surface on the rate of surface erosion under a flow-induced shear stress. A description of the experimental procedures and results from these experiments have been given by Parchure (1980), Dixit (1982) and Mehta et_al_. (1982a). A synopsis is given here. A commercial grade kaolinite with a CEC of approximately 9 meq/100 gm was used in these experiments. Tap water, with a total salt concentration of 0.28 ppt, pH = 8.5 and sodium adsorption ratio SAR = 0.012, was used in the recirculating flume, while tap water plus commercially available sodium chloride at 35 ppt concentration, pH = 8.1 and SAR = 12.0 was used in the annular flume. The kaolinite was equilibrated with the fluid for at least two weeks prior to the tests. The equilibration time is an important factor that can affect the rate of erosion due to the possibility of concentration gradients of ionic constituents between the solid and the liquid phases, or between the pore fluid and the eroding fluid, in the sediment-water system if the time allowed for equilibration is insufficient (Mehta, 1981). 85 The resuspension test methodology is depicted in Fig. 3.22. Specifically, this figure shows how the bed shear was varied over the course of each experiment. In Phase I the sediment-water mixture with sediment concentration was mixed at a high shear stress, i;^, for a period T^. The shear stress t was greater than v , the maximum bed max shear stress at which deposition of suspended sediment occurs. In Phase II the bed shear was lowered in steps, to for T^ , then to for Tj^ and finally to zero shear stress for a period of consolidation, J^^. During this phase the sediment settled out of suspension, formed a bed and began to consolidate. As indicated in Fig. 3.22, the first two phases define the pre-erosion stress history of the bed. In Phase III, the shear stress was increased as shown in discretized (one-hour) steps, i.e. = T2 = ...T^. (= 1 hour), and resuspension of the deposited material occurred. The following parameters were held constant in most of the experiments in each flume: C^, x^. T^, t^^, t^^, t^^' ^nd T^- . Shown in Fig. 3.23 are typical values for these parameters and the measured variation of suspended sediment concentration with time for a test in tap water. The parameter At^. , which may be referred to as the normalized incremental bed shear stress, is defined as (x. ^.-xAf-z. 1+1 1 1 where t. is the bed shear stress, x^, during the ith time- step. In steps i =1,...,5 in this figure it is apparent that the suspension concentration approaches a constant value during the latter stages of each time-step, i.e. dC/dt^, while for steps i = 5 and especially 7 the values of dC/dt do not approach zero over the one hour periods. This difference in the concentration-time profiles is represented in a different manner in Fig. 3.24, which shows the 86 0) £ A 8? T i rO if h- O -a CVJ CVJ a Jl 1 c g c CD Q. CO Z3 CO <D IT V- O CO CD + 1 g (/) o LU I CD Q_ 13 CO • O) CM •r- CO I— CTl 4J O fd o ^ •r- OI +-> s: ra -I- i. S- O) (O +-> QJ -C -M U I/) O O) 4- 1— >> C O O -r- I (/I OJ Q. -M ZS 2: cu o; -a OJ -a 4-) c o ra CD r— c o o -a o 00 4-5 O) ra J= S- +J ra Q. q- QJ o s- Q_ c O T3 •.- O) +J CQ ro 4- > CD C C <D •!- to S- CU 3 5- Q Q. OJ c/l +-1 ra e cu 00 s_ ra OJ 00 00 CM CVJ ro •I— u. 87 \ E CP o g < cr H u o o 00 z: a. CO Z) CO 3.50 11.00 10.00 -Tr ELAPSED TIME T (hrs) 4.50 5.50 6.50 7.50 6.00- 0.00 '•00 2.00 3.00 ELAPSED TIME T (hrs) 4.00 Fig. 3.23. Variation of Suspension Concentration with Time for T =48 Hours (after Dixit, 1982). dc 88 - CQ = 44.i gm/Iiter h =30.5 cm I Fig. 3.24. C(T^) Versus t^. for Three Values of T^^, Using Kaolinite in Salt Water (after Mehta et al. , 1982a). 89 suspension concentration at the end of step i, C(T^. ), plotted against t. for three different tests in the recirculating flume. The value of -v^^, a characteristic shear stress (similar to one defined previously for tests with placed beds) is determined as shown for each test. It is apparent that dC(T^-)/dt is higher for t^- > t^^^ than for i^ < x^^. The significance of this observation is better appreciated when it is realized that £ is proportional to the excess shear stress, and that increases more rapidly with depth, z^, for z^^ < z^,^, where z^^ is the depth below the initial bed surface at which = (Mehta et al_., 1982a). The following empirical relationship between and t:. = x^{z^) was derived from these experiments: e. = ^,_^exp[a.. 1 c b c b (3.13) where and <x. are empirical coefficients. Figures 3.25 and 3.26 show this relationship for tests in tap water and salt water, respectively. This relationship is analogous to the rate expression which results from a heuristic interpretation of the rate process theory for chemical reactions (Mehta et_ al_., 1982a). Christensen and Das (1973), Paaswell (1973) and Kelly and Gularte (1981) have used the rate process theory in explaining the erosional behavior of cohesive sediment beds. By analogy, e^- is a quantitative measure of the work done by t^- on the system, i.e. the bed, and e„ and a./i t-, \ ... ^ , ' 0^- '^■\/ '-(.[z^] are measures of the system s internal energy, i.e. bed resistance to an applied external force. 90 Fig. 3.25. Normalized Rate of Erosion, e./e.^. Versus Normalized Excess Shear Stress, , Using Kaolinite in Tap Water (after Mehta et a^. , 1982a). 91 Series 3 T(j(.= 40 hr i (g cmViin"') • 1 0.100 5.9 0.04 o 2 0.120 5.5 0.25 ^ 3 0.145 5.5 0.30 a 4 0.175 5.5 0.27 * 5 0.210 84 0.22 1 1 1 1 1.0 i.5 2.0 2.5 3.26. Normalized Rate of Erosion, e^/e^^. Versus Normalized Excess Shear Stress, {t^/t^U^)) /t^U^) , Using Kaolinite in Salt Water (after Mehta et al . , 1982a). 92 An important conclusion reached from the above experiments was that new deposits should be treated separately from settled, consolidated beds (Mehta et_ a1_. , 1982a). The rate of surface erosion of new deposits may be evaluated using Eq. 3.13, while the erosion rate for settled beds may be suitably determined using Eq. 3.12, in which £ varies linearly with the normalized excess bed shear stress. The reasons for this differentiation in determining e are twofold: 1) typical -^^ and p profiles in settled beds vary less significantly with depth than in new deposits, and may even be nearly invariant. Therefore, the value of ^\/^c^ " 1 = ^"^b w^"'"' relatively' small . For small values of At,^, the exponential function in Eq. 3.13 can be approximated by a* (1 + Atj^) which represents the first two terms in the Taylor series expansion of exp{a(ATjj)). For small values of At^, i.e. L-z^ « i, both expressions for £ vary linearly with A-c^^. Thus, the variation of £ with depth in settled beds can be just as accurately and more simply determined using Eq. 3.12. 2) The laboratory resuspension tests (briefly described in Appendix D) required to evaluate the coefficients and a for each CND layer can not be practically or easily performed using vertical sections of the original settled bed (obtained from cores). A simpler laboratory test has been described by Ariathurai and Arulanandan (1978) to evaluate the variability of M with depth. This procedure is briefly noted in Appendix D. 3.4.2. Effect of Salinity on Resuspension Sherard_et^. (1972) have shown that the susceptibility of a cohesive sediment bed to erosion depends on two factors: 1) the pore fluid composition, as characterized by the SAR, and 2) the salinity of 93 the eroding fluid. It was found that as the eroding fluid salinity decreases, soil resistivity to resuspension decreases as well. These results were verified by Arulanandan et al_. (1975). In addition, Sherard et a][. (1972) found that the erosion resistance decreased by either the exchange of cations or a reduction of the valence of the cations in the pore fluid. Kandiah (1974) and Arulanandan et al_. (1975) confirmed these findings by showing that the erosion resistance decreased and the rate of resuspension increased with increasing SAR (and therefore decreasing valency of the cations) of the pore fluid. Figure 3.27 shows such a relationship between the SAR and the critical shear stress for erosion, which is a measure of soil resistance to erosion (Alizadeh, 1974). Experiments were conducted during this study to determine the effect of the eroding fluid salinity on the rate of resuspension. The experiments were performed in the rotating annular flume using the bottom sediment from Lake Francis, Nebraska. Analysis of water from the lake indicated the presence of Na"^, K"^, Ca^"^, Mg^^, Al^"^, Fe^"*", CI" SO4 . These cations and anions would be expected to be present in the sediment as well. The average pH of the lake water was 8.6. The sediment was repeatedly washed in an attempt to remove these free salts so that their effect on the sediment properties was minimized. The washing was performed by immersing the sediment in deionized water, vigorously stirring the sediment and water, allowing time for the sediment to settle out of suspension by gravity, and then siphoning off the clear supernatant water. This procedure was repeated at least three times. 94 SODIUM ADSORPTION RATIO, SAR Fig. 3.27. Critical Shear Stress Versus SAR for Montmorillonitic Soil (after Alizadeh, 1974). 95 Commercial grade sodium chloride dissolved in different proportions in tap water constituted the eroding fluid in these experiments. The manufacturers of the sodium chloride supplied the data given in Table 3.2 regarding the contents of the processed sodium chloride. The cation concentrations in sea salt, also included in this table, were obtained from Bolz and Tuve (1976). Table 3.2 Cation Concentrations in Processed Sodium Chloride and- Standard Sea Salt Cation NaCl Sea Salt Sodium 357460. ppm 301720. ppm Calcium, Magnesium 50. 47770. Potassium 10. 10860. Phosphate 1.0 Iron 0.5 Tests were conducted for the following five salt concentrations: 0, 1, 2, 5 and 10 ppt by weight. However, as the concentrations of the three most abundant cations, Na"^, Ca^"^ and Mg^'^, in the manufactured salt were different from those in the standard sea salt (see Table 3.2), the five different eroding fluids used in these experiments were not exactly equivalent to sea water at the various salinities. In spite of this, useful qualitative and quantitative information was obtained regarding the effect of varying dissolved salt (i.e. electrolyte) concentrations on the erosive characteristic of the mud. I 96 The experimental procedures used in these tests has been described in the previous section. Suspended sediment concentration as a function of time as well as the bed density profile were measured. The values of and Tj were 0.9 N/m^, 24 hours and 40 hours respectively (see Fig. 3.22). The bed shear stress during resuspension ranged from 0.14 to 0.52 N/M'^. The bulk density and shear strength profiles for each salt concentration are shown in Figs. 3.13 and 3.14. Rates of resuspension, £, were calculated from the concentration- time profiles in the following manner. Smooth curves were drawn through the data points and values of the concentration were read off these curves at 0, 5, 15, 30 and 60 minutes after each change in the bed shear stress. Values of dC/dt were determined using a backward difference differentiating scheme. Values of e were calculated using Eq. 3.12. The logarithm of the erosion rate was plotted against the average normalized excess shear stress, i.e. {^^ - where is the average shear strength of the bed layer that was eroded by the bed shear stress i:,^. Figure 3.28 shows these plots for the 1 ppt salinity test. The slope of each line, a, and the ordinate intercept, e^, were determined from each graph. The values of a and Eq plotted as a function of depth for each salt concentration are given in Figs. 3.29 and 3.30 respectively. Before evaluating these results it is appropriate to discuss parameters other than the salt concentration that varied from test to test, in order to examine the possible significance of their variance on the rate of erosion. The other uncontrolled parameters were the temperature, pH and the SAR of the eroding fluid. Also, the rotating annular flume does not have the facility to maintain a constant water 97 Z Tm =0.9N/m^ - Tm =24 hrs. 0 0.05 0.1 0.15 0.2 (^) Fig. 3.28. Resuspension Rate Versus Normalized Excess Shear Stress, 98 £ 0 2 4 6 8 10 . 3.29. Slope, a, Versus Depth Below Bed Surface, z, , as a Function of Salinity. ° 3.30. Ordinate Intercept, e^. Versus Depth Below Bed Surface, Zj^, as a Function of Salinity. 99 temperature during the course of an experiment. As a result the temperature typically varied 3° to 5°C over the seven to eight hour duration. A temperature variation of this magnitude has been found to result in less than a 3% decrease in the bed shear strength (Kelly and Gularte, 1981) and is considered to be insignificant. Likewise, over the one per-cent salt concentration range used in these experiments the variation in pH is considered to be not significant. However, due to the relatively small quantities of Ca^"*" and Mg^"*" compared to that of Na"*", the SAR values were rather large and increased significantly with increasing salt concentration. For example, the SAR values varied from 110 to 349 as the salinity increased from 1 to 10 ppt. Alizadeh (1974) showed that both the concentration of the electrolyte and the SAR are important controlling factors in the process of coagulation. Specifically he found that the effect of salt concentration gradually decreases with increasing SAR. Thus, the varying SAR values are considered to have had some, albeit unmeasured, effect on these experiments. Analysis of the variation in the bed density profiles with salinity, shown in Fig. 3.13, revealed no discernable relationship. It is felt that further investigations are necessary to determine if any relationship exists between Pg and the salt concentration of the eroding fluid for a stratified cohesive sediment bed. The bed shear strength profiles, shown in Fig. 3.14, were analyzed by determining the weighted depth-averaged value (weighted with respect to spacing, i.e. depth, between adjacent data points) of at the five different salt concentrations, S. The following relationship was found: i 100 -^^(S) = T^(S=0)*(S/2 + 1) for 0 £ S < 2 (3.14) -^^(3) = 2t^(S=0) f or S >_ 2 where S is in ppt. This relationship was incorporated into the bed formation algorithm in the following manner. The discretized value of at the top of the uppermost new deposit bed layer is changed instantaneously, i.e. during the same time-step, at every element where the elemental average salinity value changes. For the second bed layer the discretized value is not changed during the first time-step during which the average value of salinity changes; it is changed at the time-step during which the salinity changes for the second time. However, for this bed layer the new value is determined using Eq. 3.14 and the second preceding value of the salinity at that element. This procedure is similarly repeated for the remaining new deposit bed layers. This method of incorporating the effect of the salinity of the eroding fluid on the bed shear strength profile was used in order to account, at least partially, for the finite amount of time it takes for denser (i.,e. higher salinity) eroding fluid to diffuse downward into the bed or for denser pore fluid to diffuse upward into the overlying eroding fluid. The diffusion coefficients of CI" and Na"^ in Pacific red clay and Lake Ontario sediment were experimentally determined to be of the order of 10"^ - 10"^ cm^/sec at a temperature of 24°C (Li and Gregory, 1974; Lerman and Weiler, 1970). These extremely small diffusion coefficients indicate that the rates of diffusion in unconsolidated sediments are generally from one half to one twentieth of the diffusion rates in the eroding fluid (Manheim, 1970). 101 For the first time-step the initial salinity value at each element is used to determine the values in both the unconsolidated and the partially consolidated bed layers, while the salinity of the pore water in the original settled bed layers, an input parameter in the model, is used to evaluate the values in this bed section. The i^ values of the settled bed layers are thereafter assumed to be invariant with respect to the salinity of the eroding fluid. The justification for this assumption is based on the observation that dissolved silica concentrations in pore waters of Lake Ontario, Erie and Superior sediments were, in general, invariant with respect to depth after the first 20 cm below the water-mud interface (Nriagu, 1978). Therefore, the salinity of the eroding fluid would not be expected to influence that of the pore fluid below the top 20 cm of the bed, which clearly encompasses the consolidated bed section. The values of a and are seen in Figs. 3.29 and 3.30, to decrease and increase, respectively, with increasing salinity. However, inasmuch as these parameters are considered to be characteristic properties of the sediment bed, and as the effect of salinity on another bed property, -c^, which is as well estimated indirectly from measured data, has already been incorporated into the model, it was not necessary to consider the variation of a and with salinity in the erosion algorithm. 3.4.3. Erosion Algorithm A description of the redispersion and resuspension algorithms is given below. In both algorithms, the rate of erosion is calculated on an element by element basis. 102 A portion of the unconsolidated new deposit (UND), when present, will redisperse (mass erode) when is greater than the surface shear strength of the UNO, i.e. -^^{2^=0). The thickness of the UND that fails totally and is instantly redispersed is equal to z^^, where z^^ = bed depth at which ■^^(25) = "^5- The value of z^^^ is determined from the linearly varying '^f.iz^) profile in each UND layer. The value of z^^ may be greater than the thickness of the top layer, TLAYM(l), in which case more than one layer is redispersed. The dry mass of sediment that is redispersed, Mp, is calculated according to = J P(z )dz (3.15) 0 where Mq has units of Kg/m'^ and is considered to be the mass eroded over one time-step At. The contribution to the source term in the governing equation (Eq. 3.5) caused by redispersion is given by Eq. 3.15 divided by the product of the average elemental water depth and the time-step At. New UND layer(s) thicknesses and '^f^iz^) and p{z^) profiles are calculated at each time-step when redispersion occurs by subtracting z^ and resetting T:^iz^=0) and p(Z|^=0) equal to the respective initial values at z^ = z^^. If z^^ is calculated to be greater than the thickness of the entire UND, then all of this sediment is redispersed. For both the redispersion and resuspension algorithms, erosion is considered to occur only during accelerating flows, i.e. Tj^(t+At) > \it). Thus, even though ^{^(t+At) may be greater than x^{z^=0), no erosion will occur if > ^^(t+At). This stipulation for the occurrence of erosion, and an analogous one for deposition (as will be discussed in Section 3.6), is based on an interpretation of the 103 typically observed Eulerian time-concentration variation in an estuarial environment. For example. Fig. 1.3 shows a time-concentration profile from the Savannah River estuary (Krone, 1972). Also indicated is the observed correlation between accelerating flows and increasing suspension concentration and between decelerating flows and decreasing suspension concentration. Laboratory evidence (Mehta and Partheniades, 1975; Partheniades, 1977; Mehta et^., 1982a; Parchure, 1983) suggests that under accelerating flows, erosion occurs without redeposition of the eroded sediment. Likewise, during decelerating flows, sediment deposits without reentrainment of the deposit. During periods of steady flows, erosion or deposition may occur. These two processes do not, however, occur simultaneously even in this case (Parchure, 1983). The initial condition at the inception of the steady flow period will determine whether erosion or deposition will occur. If the antecedent phase was one of acceleration, the sediment will continue to erode under the steady flow condition. In both cases, however, relatively short transient periods of simultaneous erosion and deposition sometimes do tend to occur (Yeh, 1979). For estuarial modeling purposes, however, these periods may be ignored without introducing any significant errors. Resuspension of partially consolidated beds (CND) occurs when: 1) the entire UND has been redispersed, 2) T|^(t+At) > "^^it) and 3) 'i:|^(t+At) > -^^(21^=0), where z^=0 is now at the fluid-CMD interface. The resuspension rate expression (Eq. 3.13) found by Mehta _et al_. (1982a) is used to determine the thickness of the CND, zu , that is resuspended during a time-step. At. The following iterative procedure is used to calculate z^^ during any given time-step. 104 The average erosion rate, e, for the period At is calculated as: I =V2 [£(t) + e(t+At)] (3.16) in which (t+At) D e(t+At) = e^CDexpr (1)( -1)] (3.17) c where Sgd) and a(i) are the average empirical coefficients for the first (i.e. top) CMD layer, and 1 2. \ =^^2!:\(Zb=0) + — / ^*-^^(Zj^)dz] (3.18) As a first guess, z^^ is set equal to TLAY(l) = z^^_^ (see Fig. 3.15). A new value of Zj^ , designated Zj^ , is calculated according to: * *2 (3.19) where p is the average dry bed density over the first bed depth Z[^^_^. Then the following parameter is evaluated: P ^b ^ - 1 = \K-\\ (3.20) A\ = _ 1 = \x-l\ sAt 105 where p and e are determined using z = z . If aa. < 0.02, then Zu is taken to be the depth of bed eroded during this time-step. If AA. > 0.02, then yet another new value of z^^, designated is calculated using the following equation: =z + 1 (3.21) where = P'zu /(e*At^' '^''"'^^ ^ ^""^ ^ determined using Zj, = z^ . ^^■'"9 ^b^vo' ^'^^ entire procedure, i.e. Eqs. 3.16 through 3.21, is *3 repeated until the chosen error criterion, i.e. aa < 0.02, is satisfied. As in the redispersion routine, new CND layer thickness(es) and and p(z^) profiles are determined. As before, z^^ may be greater than the thickness on the top layer. Laboratory tests required to evaluate i^c^^b^' *^^^b^» ^""^ average values of and a for each CMD layer are described in Appendix D, Section D.2. Once the entire new deposit bed section has been eroded, the original settled bed, if any exists, will undergo resuspension when the following two conditions occur: 1) t|^(t+At) > \{\.) and 2) T[^(t+At) > 'Cj,(Zfj=0), where Zfj=0 is now at the top of the settled bed. The surface erosion rate expression (Eq. 3.12) given by Ariathurai and Arulanandan (1978) is used to evaluate the thickness, z^,^, of the settled bed that is eroded during each time-step. The iterative procedure used for the CND is again used to solve for Zj,^, with only the expression for £ being different,. Equation 3.16 becomes 106 e(t+At) = M(l)( : 1) (3.22) c where M(l) is the erodibility constant for the first layer. The contribution to the source term in Eq. 3.5 caused by resuspension is given by Eq. 3.16, with Eq. 3.17 used for the partially consolidated bed section and Eq. 3.22 used for the original settled bed section, divided by the average elemental water depth. In the following section, the dispersive transport of suspended sediments is discussed, followed by a description of the dispersion algorithm. 3.5. Dispersive Transport 3.5.1. Dispersion Mechanisms There have been numerous studies on the dispersion of some quantity (e.g. sediment) in a bounded shear flow in the thirty years since the work of Taylor (1953, 1954). Taylor proved that a one-dimensional dispersion equation can be used to represent the longitudinal dispersion of a quantity in turbulent pipe flow. Taylor's analysis has since been extended to shear flow in both rivers and estuaries. The present discussion is limited to dispersion in estuary flow. A brief review of dispersive transport theory precedes that of estuarial dispersion. The governing equation (Eq. 3.5) derived for the two-dimensional, depth-averaged movement of suspended sediment in a turbulent flow field includes dispersive transport terms which account for the transport of sediment by processes other than advective transport. Some of these other processes include the effects of spatial (i.e. transverse and vertical) velocity variations in bounded shear flows and turbulent 107 diffusion. Thus, the effective sediment dispersion coefficients in Eq. 3.5 must include the effect of all processes whose scale is less than the grid size of the model or what has been averaged over time and/or space (Fischer et__al_. , 1979). For example, the effect of the actual vertical concentration gradient would have to be incorporated in the dispersion coefficients in the present depth-averaged transport model. Diffusion is defined as "the transport in a given direction at a point in the flow due to the difference between the true advection in that direction and the time average of the advection in that direction," and dispersion is defined as "the transport in a given direction due to the difference between the true advection in that direction and the spatial average of the advection in that direction" (Holley, 1969). Holley enunciates the fact that diffusion and dispersion are both actually advective transport mechanisms, and that in a given flow field, the relative importance of one mechanism over the other depends on the magnitude of the concentration gradient in the particular transport problem. In the governing equation (Eq. 3.5) the effective sediment dispersion coefficients are equal to the sum of the turbulent diffusion coefficients and dispersion coefficients. This approach follows the analysis of Aris (1956) which showed that the coefficients due to turbulent diffusion and shear flow (dispersion) were additive. Thus, the analytic expressions to be used for the effective sediment dispersion tensor would include, at least in some sense, both diffusion and dispersion. Fischer (1966) showed that the dispersion of a given quantity of tracer injected into a natural stream is divided into two separate phases. The first is the convective period in which the tracer mixes 108 vertically, laterally and longitudinally until it is completely distributed across the stream. The second phase is the diffusive period during which the lateral and possibly the vertical (depending on the nature of the tracer) concentration gradient is small, and the longitudinal concentration profile is highly skewed. The governing equation (Eq. 3.5) is strictly valid only in the diffusive period. The criterion for determining if the dispersing tracer is in the diffusive period is if it has been in the flow longer than the Lagrangian time scale and has spread over a wider distance than the Lagrangian length scale (Fischer et_ al_., 1979). The latter scale is a measure of the distance a particle travels before it forgets its initial conditions (i.e. position and velocity). Analytic expressions for the sediment (mass) diffusion coefficients can be obtained by analogy with the kinematic eddy viscosity. Specifically, the Reynolds analogy assumes that the processes of momentum and mass transfer are similar, and that the turbulent diffusion coefficient, E, and the kinematic eddy viscosity, e^, are in fact linearly proportional. Jobson and Sayre (1970) verified the Reynolds analogy for sediment particles in the Stokes range (less than about 100 m in diameter). They found that the "portion of the turbulent mass transfer coefficient for sediment particles which is directly attributable to tangential components of turbulent velocity fluctuations: (a) is approximately proportional to the momentum transfer coefficient and the proportionality constant is less than or equal to 1; and (b) decreases with increasing particle size." Therefore, the effective sediment mass dispersion coefficients for cohesive sediments may be justifiably assumed to be equal to those for the flow itself. 109 Fischer ^ a]_. (1979) define four primary mechanisms of dispersion in estuaries: 1) gravitational circulation, 2) shear-flow dispersion, 3) bathymetry induced dispersion and 4) wind-induced circulations. These four mechanisms are briefly described next. Gravitational or baroclinic circulation in estuaries is the flow induced by the density difference between the fresh water at the landward end of the estuary and the sea water at the ocean end. There are two types of gravitational circulation. Transverse gravitational circulation is depth-averaged flow that is predominantly seaward in the shallow regions of a cross-section and landward in the deeper parts. Figure 3.31a depicts this net depth-averaged upstream (landward) and downstream (seaward) transport and the resulting transverse flow from the deeper to the shallower parts of the cross-section. Fischer et al . (1979) state that "the upstream flow is expected to be concentrated in the deeper portions of the channel, because the upstream pressure gradient increases linearly with depth below the water surface." Thus, the interaction between the cross-sectional bathymetry and the baroclinic flow causes the transverse circulation. Vertical gravitational circulation is schematically illustrated in Fig. 3.31b which shows the predominantly seaward flow in the upper part of the flow and landward flow in the lower part. Fischer (1972) believes that the vertical gravitational circulation will be more important than transverse circulation only in highly stratified estuaries. The previously described mechanism of shear-flow dispersion is believed to be the dominant mechanism in long, fairly uniform sections of well-mixed and partially stratified estuaries (Fischer et al.. 110 (a) NET DOWNSTREAM TRANSPORT "NEAR THE SURFACE VECTICAL AOVECTION AND DIFFUSION Fig. 3.31, NET UPSTREAM TRANSPORT NEAR THE BOTTOM (b) The Internal Circulation Driven by the River Discharge in a Partially Stratified Estuary, (a) A Transverse Section; (b) A Vertical Section (after Fischer et al_. , 1979). Ill 1979). Holley et aT_. (1970) applied the dispersion analysis of Taylor (1954) to oscillating flow in estuaries. They concluded that for wide estuaries, the effect of the vertical velocity distribution on shear- flow dispersion is dominant over that of the transverse velocity distribution. The exact opposite situation was found for relatively narrow estuaries. The joint influence of bathymetry and density differences on dispersion has already been mentioned in the discussion on baroclinic circulation. Other examples of bathymetry induced dispersion include: the intrusion of salinity or sediment in certain parts of a cross- section caused by the channelization of flood and ebb tides in tidal inlets or narrow estuaries (Fischer etal_., 1979); and the enhanced dispersion of a quantity (e.g. pollutant) or intrusion of salinity in tidal flats and side embayments, which serve as storage areas for these constituents, caused by the out of phase flow which occurs between the main channel and such features (Okubo, 1973). An example of a wind-induced circulation is shown in Fig. 3.32. The steady onshore wind causes a circulation in the wind direction in the shallow bay, where the less water mass per unit surface area results in a higher acceleration and therefore quicker response to the wind- induced surface stress, and in the opposite direction in the deeper sections of the channel. Such a circulation can cause significant dispersion (Fischer _et_al_. , 1979). For a detailed description of dispersion, the reader is referred to the following references: Dispersion in estuaries - Glenne and Selleck (1969), Pritchard (1969), Holley (1969), Holley etal_. (1970), Fischer (1972), Okubo (1973), Ward (1976), Fischer (1978), Murray and Siripong 112 (1978), Zimmerman (1978) and Fischer ^al_. (1979); Dispersion in channel flows - Fischer (1966), Dagan (1969), Fischer (1970), Peterson et al_. (1974), Taylor (1974), Ward (1974), Sumer and Fischer (1977), Smith (1978), Beltaos (1980a, b), Chatwin (1980) and Liu and Cheng (1980); Dispersion of particle matter - Sayre (1969), Jobson and Sayre (1970), Chen (1971), Sumer (1971) and Alonso (1981). 3.5.2. Dispersion Algorithm The most important, and possibly the most difficult task in modeling dispersion is to determine which of the dispersion mechanisms are important in the estuary being modeled. For example, if the estuary has only a few tidal flats and shore irregularities and has a fairly uniform cross-section (e.g. the Delaware River), shear flow dispersion may be the dominant mechanism. However, if the estuary is relatively deep and the river discharge is large (e.g. the Mississippi River), gravitational circulation may be just as or even more important than shear flow dispersion. Unfortunately, none of the existing dispersion models, most of which are two-dimensional (e.g. DISPER (Leimkuhler _et_ al . , 1975)), can represent the combined effects of, for example, an irregular shoreline configuration and bathymetry, shear flow dispersion and baroclinic flow. Because of these problems in identifying, describing and modeling the various dispersion mechanisms which occur in estuaries, the decision was made to develop a dispersion algorithm for only shear flow dispersion that would be applicable to a wide, vertically well mixed estuary. Following the analysis of Hoi ley et al . (1970), it is assumed that the dispersion in wide estuaries is associated primarily with the 113 vertical shear. The limitations, which determine the applicability of such a dispersion algorithm, are consistent with those associated with a two-dimensional, depth-averaged cohesive sediment transport model. The dispersion tensor derived by Fischer (1978) for two- dimensional, depth-averaged bounded shear flow is used in the dispersion algorithm. The four components of this tensor are D^y = (UVd2/E)I^y (3.23) Dyj^ = (UVd2/E)Iy^ Dyy = (V2d2/E)lyy I in which: U and V are the root-mean-square values of u and V over the depth d; u' = u(z) - u, where u is the depth-averaged component of the velocity in the x- di recti on; V = v(z) - V, where v is the depth-averaged component of the velocity in the y- di recti on; E = mean value of the scalar turbulent diffusion coefficient in the vertical direction, E^; and 1 „ CI C „ I. . = Ju. / — Ju.dCdCdC (3.24) iJ 0 ^ 0 E' 0 114 in which: E = E^/E, u^- = u^-/u and C = z/d. The quantities u and v are the velocity deviations taken over the depth from the respective depth-averaged values, u and v. The values U and V represent the I I "intensity" of u and v , respectively (Fischer et_al_., 1979). The physical interpretation of the cross product dispersion coefficients D^^y and Dy^ is that a velocity gradient in the direction can produce mass (dispersive) transport in the Xj direction. Fischer (1978) notes that since in most investigations the vertical velocity profile, i.e. u(z) and v(z), and the vertical turbulent diffusion coefficient, E^, are not known with a high degree of accuracy, it would usually suffice to assume that the value of I^-j in Eq. 3.24 is a constant. The value of I^-j in various parallel shear flows ranges from 0.054 for turbulent pipe flow to 0.10 for laminar flow with a linear velocity profile over d (Fischer et_al_. , 1979). Therefore, Fischer recommends that a value of 0.10 be used for I^-j in Eq. 3.23. The following expression for E^, derived by Elder (1959) for flow down an infinitely wide inclined plane, is used in this analysis: E = Ku.zd— ) (3.25) ^ ^ d where < = von Karman turbulence constant, and u^ = shear velocity. Therefore, E is given by — Id <u d E = - / E dz = — — = 0.067u^d d 0 ^ 6 ^ (3.26) 115 with K = 0.40. The values of < obtained by Gust (1976) from the slopes of measured clay suspension velocity profiles varied between 0.3 and 0.4. Gust considered this variation a result of the mean flow experimental error of S% and not due to the presence of suspended cohesive sediments. Therefore he assumed < = 0.40 in his analysis. This is the justification for using this value of ^ in this study. Fischer (1966) found that in both laboratory experiments and in real streams that the mean value of U^/u^ was equal to 0.2. Substituting this value, and I^- ■ = 0.1 and Eq. 3.26 into Eq. 3.23 gives XX 0.2(u)^d^ (u)^d D = (0.10) = 0.30- n.067u^d u^ 0.2uvd^ uvd D = (0.10) = 0.30 0.067u^d f- "f 0.2vud ^ uvd -(0.10) = 0.30 (3.27) 0.067u^d u^ 0.2(v)^d^ (v)^d D = (0.10) = 0.30 yy 0.067u^d u^ These are the coefficients used in the dispersion algorithm to model the shear flow dispersion of suspended cohesive sediments in a wide, well mixed estuary. Values of D^.j are calculated at each time step in the model using the specified nodal values of u, v and d. 116 In the following section, the depositional behavior of cohesive sediments is summarized, followed by a description of the deposition algorithm, 3.6. Deposition 3.6.1. Previous Investigations Deposition has been defined to occur when t:^ is not high enough to resuspend sediment material that settles onto and bonds with the bed surface. This process, therefore, involves two other processes, settling and bonding, i.e. cohesion.' Laboratory studies on the depositional behavior of cohesive sediment in steady turbulent flows have been conducted by, among others. Krone (1962), Rosillon and Volkenborn (1964), Partheniades (1965), Partheniades et al_. (1966), Migniot (1968), Lee (1974), Mehta and Partheniades (1975) and Mehta et al . (1982b). The results from these and other studies on the settling rates of cohesive sediments pertinent to the deposition algorithm described in Section 3.6.4 are summarized below. In laboratory flumes, the depositional behavior is usually investigated by allowing sediment suspended in a flume at a high shear stress to deposit by reducing the shear stress. Since the sediment concentration gradient in the direction of flow in usually small, the observed time-rate of change of the depth-averaged concentration, C, is due to the deposition of suspended material. The conservation of sediment mass can be expressed as (Einstein and Krone, 1962): .r (OC dC d s dt (3.28) 117 where t = time, d = flow depth, WgCO = sediment settling velocity as a function of C, and = probability of deposition, or the probability of a sediment particle or floe bonding to the bed and not being instantly resuspended. Krone (1962) conducted a series of depositional tests in a 31 m long and 0.90 m wide recirculating flume using mud from San Francisco Bay which contained approximately equal proportions of clay and silt. Krone postulated that P^ increases linearly with a decrease in according to P . = 1 (3.29) cd where = critical shear stress for deposition, above which no deposition occurs. Therefore, P^ decreases linearly from a value of 1.0 at T^f, = 0, to 0 for > x^^. The value of was found to be equal to 0.06 N/m^ for the Bay mud with C < 0.3 g/1. Krone found that when C < 0.3 g/1, Wg was independent of C. In this case, integration of Eq. 3.29 gives C d s — = exp[ 1] (3.30) C„ d 0 where Cq is the initial suspended sediment concentration. Thus, according to this equation all the suspended sediment will eventually deposit when '^b < t j. 118 For 0,3 g/1 < C < 10 g/1 and for C > 10 g/1 , logarithmic laws of the following form were derived: log C = -K[log(t)] + Constant (3.31) where K was found to be a function of d and P^^. Krone attributed the variation of the depositional properties with suspension concentration to different forms of settling. Various forms of settling of coagulated cohesive sediments are discussed later in this section. Partheniades (1965) conducted deposition tests in a open, flow recirculating flume using Bay mud. He noted that for flows above a certain critical bed shear, the suspended sediment concentration, after an initial period of rapid deposition, approached a constant value, which he referred to as an equilibrium concentration, Cg^. The ratio C^„/C- = C* was found to be a constant for given flow conditions, regardless of the value of Cg. Whereas for bed shears even slightly less than this critical value, all the sediment eventually deposited. Partheniades et_ al_. (1966) conducted deposition experiments in a rotating annular flume (similar to the one at the University of Florida, but with mean diameter of 0.82 m and 0.19 m wide) at the Massachusetts Institute of Technology using a commercial grade kaolinite. Based upon these experiments it was concluded that Cg^ represents the amount of sediment that, having settled to the near bed region, cannot withstand the high shear stresses present there (due to insufficiently strong interparticle bonds) and are broken up and resuspended. In addition, Cgq in the fine sediment deposition tests appears not to be the result of an interchange between suspended and bed material as it is for 119 cohesionless sediment, because if such were the case, Cg^ would not be dependent on C^. Therefore, it follows that C^^ does not represent the maximum sediment carrying capacity of the flow, as it does in the case of cohesionless sediment, but instead may be considered to be the steady state concentration (Mehta and Partheniades, 1973). As noted by Mehta and Partheniades (1975), Krone did not observe Cgq in his tests because most of them were conducted at < i^j,^, wherein C would be expected to be equal to zero. It is apparent that the definition of ?^ must be extended to include bed shear stresses greater than t^^. Mehta and Partheniades (1975) investigated the depositional properties of a commercial grade kaolinite in distilled water and in salt water at seawater salinity (35 ppt) in a rotating annular flume facility at the University of Florida. Figure 3.33 shows typical suspended sediment concentration-time plots found in these tests. It is evident that a steady state concentration was reached in each test and that for bed shears above approximately 0.16 N/m^, the value of Cgq was greater than zero and in fact increased monotonically with increasing t:^. Figure 3.34 shows the ratio C*q = Cgq/Cg plotted against for all the tests with kaolinite in distilled water. Two important conclusions are obtained from this figure: 1) C*q is a constant for a given (and type of sediment) and is not a function of depth, d, or C^, and 2) for "^s < . C* = 0. The first conclusion is based on the ° "mi n ^" observation that the data points for all the different flow conditions are almost randomly scattered about a "best fit" line. The minimum bed shear tu » observed in Fig. 3.34 is the same as the \d value defined by Krone (1962), and the critical bed shear obtained by Partheniades 120 Cr-^ — ^ ^odcidcidcic X d — o> irt (Ti (o d't oOOOOOO •dddddddo 0) n3 OJ 00 Q C +-> o S- o 4- 0) (/I CU o -o o +J fa OJ s- 121 (1965). As observed in this figure, found to be approximately '^min 0.18 N/m^ for kaolinite in distilled water. In Fig. 3.35 the data of Fig. 3.34 are plotted on log-normal coordinates as C^^ in percent against where \ = ^b'^'^b^^-^- straight line through the data points gives the following relationship between these two dimensionless parameters: C =1/2 (1 + erf (— )) eq /2 (3.32) with ^a = ^'ho^ (3.33) where is the standard deviation and (v^^50 geometric mean of the log-normal relationship given by Eq. 3.32, and erf is the error function (Mehta and Partheniades, 1975). It was found that for all tests with kaolinite in both distilled and salt water, with a 50/50 mixture of kaolinite and San Francisco Bay mud in salt water, with only Bay mud in salt water, and for the reanalyzed deposition tests of Rosillon and Volkenborn (1964), Partheniades (1965) and Partheniades et__al_. (1968), the value of was 0.49. Therefore, as noted from Eq. 3.33, C*^ is dependent solely on the value of the ratio (Vl)/{V^^50- '^^^^^ ^"^ Partheniades ( 1973) found for deposition tests in salt water the following relationship between Xin ^""^ ^Vl)50: 122 123 = 4 exp(-1.27 _ ) (3.34) mi n Mehta and Partheniades (1975) found the following dimensionless log-normal relationship for the variation, i.e. decrease, of the suspended sediment concentration v/ith time: ^ T C =1/2 (1 + erf( ) (3.35) /2 where T = logjQ(t/t5g) ^ (3.36) and where: C* = (Co'C)/(Co-Cgq) represents the fraction of the depositable sediment, C^-Cgq, deposited at any given time t, cr^ is the standard deviation of the log-normal relationship, and tgg is the geometric mean (i.e. the time at which C* = 50%). Figure 3.36 shows a comparison between some typical depositional data for kaolinite in distilled water and the log-normal relationship given by Eq. 3.35. This •k relationship was found to hold for all values of greater than approximately 0.25, with the exception that for very high Cg values (around 20-25 g/1 ) with i;^ < 1, an acceptable agreement with the measured data was not obtained. Good agreement was as well obtained between Eq. 3.35 and the data sets mentioned previously in this section. Taking the derivative of Eq. 3.35 with respect to time gives the following expression for the rate of change of C* y^ith time: 124 O B rO CM a U3 f*) cn 03 t- d d ( ) c c O If) in — d. <u d d d m o O g 4) ■o C <u <u o o a o £ o • o CD l , t: r:7_.,n:_L K -^--j — J ^ : — ( -a c to 14- n3 S- -p n3 +-> o n3 o <T>CO ID O OOOOOOO <n <T> <Ti (J) (X) f~ ^ f <^ 7o Ul ^3 c\J — O o -p CO LO 0) cn +j - c: 00 oi 03 u -a S- (0 cu •.- a. c OJ c: j= •I- +-> s- ■X (B o a. cn 125 dC* 0.434 exp(-T^/2) = (3.37) dt /2i t The standard deviation, and the geometric mean, t^g, were found to be functions of X]^, d, and C^. Shown in Figs. 3.37 and 3.38 are examples of the relationships found between these parameters. The following conclusions were arrived at from these and other similar plots given by Mehta (1973): 1) for a specific value of t^, the deposition rate was minimum. This tj^ value was found to vary between 1 and 2 for kaolinite in distilled water. The rate of deposition increased for values both less than and higher than this specific value, but not as significantly for higher values as for the lower values. However, for ^b » deposition of suspended sediment occurred. For Bay mud max in sea water, Xu was determined to be 1.69 N/m'^. 2) for x. < 1, the ■^max rate of deposition increased with an increase in d, while for Xj^ > i, the effect of d on the deposition rate was minimal. As noted, the settling velocity of suspended cohesive sediment particles has been found to be a function of, among other parameters, the suspension concentration (Krone, 1962). There appears to be at least three types of settling: 1) no mutual interference, 2) mutual interference and 3) hindered settling. For very low suspension concentrations, on the order of 0.1-0.7 g/1 , the aggregates or elementary particles settle independently without much mutual interference, and therefore the settling velocity is independent of C. For concentrations between approximately 0.3 g/1 and 10-15 g/1, the settling velocity increases with concentration due to the accompanying increase in interparticle (floe) collisions, and therefore increased mutual interference (Fig. 3.39). For concentrations higher than 10-15 126 Fig. 3.37. Log t^Q Versus for Kaolinite in Distilled Water (after Mehta and Partheniades, 1975). Fig. 3.38. Versus for Kaolinite in Distilled Water (after Mehta and Partheniades, 1975). 127 C \ I I I I I 1 I I I 0.1 0.2 0.4 0.6 0.8 1.0 2.0 SUSPENDED SEDIMENT CONCENTRATION, C(g/^) Fig. 3.39. Settling Velocity, W^, Versus Suspended Sediment Concentration, C, for San Francisco Bay Mud (after Krone, 1962). 128 g/1, the settling velocity actually decreases with increasing concentration (Figs. 3.40 and 3.41). At such high concentrations the sediment suspension, referred to as fluid mud or mud cake (Bellessort, 1973), hinders the upward flux of water expelled by consolidation of the lower suspension (Krone, 1962). In the mutual interference range. Krone (1962) and Owen (1971) have found the following empirical relationship between the median settling velocity, Wg, and C: = Kc" • (3.38) where K and n are the empirical constants that depend on the sediment type and the turbulence intensity of the suspending fluid. Krone found n to be equal to 1.33 for Bay mud in laboratory experiments (see Fig. 3.39). Teeter (1983) found n to be less than 1.0 for sediment from Atchafalaya Bay, Louisiana. Owen (1971) studied the effect of turbulence on the settling velocities of natural mud. No description of the sediment was reported, except that it was collected in the Thames River near Dagenham, England. A specially designed sampling instrument was used to collect sediment samples during both a spring tide and a neap tide. This tube collects undisturbed samples of suspended sediment in an estuarine environment, and immediately thereafter the median settling velocity of "natural aggregates" can be determined using the bottom withdrawal method described by Owen (1970). The value of n determined using this method was 1.1 and 2.2 for sediment collected during a spring and a neap tide, respectively. The turbulence intensity during a spring tide is 129 9 3.0 X in £ E o o _l > I- I- u C/0 SUSPENDED C0NCENTRAT10N,C(g/i) Fig. 3.40. Settling Velocity, W^, Versus Suspended Sediment Concentration, C, for Yangtze River Estuary Mud (after Huang et al . , 1980). Fig. 3.41, i;; 100.0 £ £ > I- UJ > UJ CO 10.0 .0 0. O.Oi- < S 0.0 i O.IO Severn estuary mud (saline water) 1.0 10.0 100.0 SUSPENDED SEDIMENT CONCENTRATION, C (g/^) Versus C for Severn Estuary Mud (after Thorn, 1981) 130 greater than during a neap tide. Owen therefore postulated that n was greater (and therefore as well) during the neap tide because the lower level of turbulence did not cause significant breakage of the aggregates; thus relatively large aggregates with higher settling velocities were formed. During the spring tide the higher degree of turbulence, and therefore greater internal shearing rates, did result in breakage of a significant proportion of the aggregates. Thus, small aggregates with lower settling rates, and therefore lower values of n, were formed. Owen believed that the interparticle collision rate was significantly high during both tides and therefore did not consider it very probable that the aggregate size would have been affected (i.e. limited) by this factor. Owen also performed standard settling tests in a one meter high bottom withdrawal tube using the same sediment samples as above, and found that varied linearly with C (i.e. n==1.0) for both spring and neap tide samples and that the values of were approximately one order of magnitude smaller than the values determined with the aggregate collection tube. The latter result is very significant in that it reveals the apparent effect of turbulence on the behavior (e.g. settling velocities) of sediment aggregates: larger, stronger aggregates with corresponding higher settling velocities are formed in a turbulent flow field than under quiescent conditions primarily because of increased collision rates due to high internal shearing rates. Migniot (1968) defined a "flocculation factor" F, given below, in order to quantify the effect of the aggregation intensity on Wg: 131 W F = (3.39) W where W- is the median settling velocity of the aggregates and W is A the median settling velocity of the elementary sediment particles. Bellessort (1973) reported that F varies with the grain size of the elementary (i.e. deflocculated) particles according to F = a^.D = (3.40) where D is the mean diameter of the particles in microns (10~^m), oc^ = 250 and = 0.9, provided is measured in mm/s. Figure 3.42 shows this effect of the particle size on F and W. for numerous sediment samples at = 10 g/1 and salinity S = 30 ppt. Also plotted in this figure is the variation of F with D found by Dixit _eta][. (1982) using mud from Atchafalaya Bay, Louisiana. However, in these data Cg varied from 1.2 to 11 g/1 as indicated and S=0.0 ppt. Another important difference between the two data sets is that Bellessort measured W. ^A under quiescent conditions, while Dixit _et__al_. (1982) measured this settling velocity under turbulent flow in the rotating annular flume. As observed, these data have the same slope between F and D as given by Bellessort, This suggests that, in general, F may be proportional to D albeit with different intercept values, at least for suspension concentrations with C- = 1.2-11 g/1 and 0<S<30 ppt. 132 MEAN DIAMETER (Microns) VELOCITY (mm/s) Fig. 3.42. Effect of Size and Settling Velocity of Elementary Particles on the Coagulation Factor of Natural Muds after Bellessort, 1973). 133 3.6.2. Effect of Salinity on Deposition The larger, stronger aggregates of natural muds formed in a saline medium have been found to result in higher settling velocities (Krone, 1962; Owen, 1970), which result in higher rates of deposition. Thus, the effect of salinity on the deposition of cohesive sediments may be quantified in terms of a relationship between salinity and the median settling velocity, Wg, of a particular sediment. Krone (1962) studied the effect of salinity and suspended sediment concentration on of sediment from Mare Island Strait in San Francisco Bay. Hydrometer analysis showed that 60% by weight of this sediment was in the clay size range (i.e. < 2 urn), with the remainder in the silt size range. X-ray diffraction and differential thermal analyses of the clay fraction revealed a large content of illite, montmorillonite and kaolinite clay groups along with small quantities of chlorite and quartz. The results from settling tests performed under quiescent conditions in one-liter cylinders showed the effect of both salinity and suspension concentration on (Fig. 3.43). The influence of salinity on is especially significant in the range 0-2 ppt, particularly for the 1.0 and 0.53 g/1 suspension concentrations. This result is expected considering the discussion presented in Section 2.2.6. One possible explanation for the apparent increasing influence of salinity on with increasing suspension concentration, as shown in this figure, is the following. As the suspension concentration increases, the number of collisions (by Brownian motion and differential settling mechanisms in such an experiment) would likewise increase and therefore promote the formation of larger aggregates with higher settling velocities. The lowest order aggregate that could be formed would be limited by the i 134 135 suspension concentration, so that even with an increase in salinity (and therefore a corresponding increase in cohesive forces), lower order aggregates with typically higher settling velocities could not form due to the insufficient concentration of suspended particles. Owen (1970) studied the variation of of a natural mud with salinity and suspension concentration. Approximately 55% of the mud was in the clay size range, with the remainder in the silt range. It was revealed that the clay fraction was composed of, in order of abundance, illite, kaolinite, montmoril lonite and chlorite. Settling tests were conducted in a two meter high bottom withdrawal settling tube. The results of Owen's tests are shown in Figs. 3.44 and 3.45. These figures show that, in general, as the salinity and suspension concentrations are increased, increased cohesion and interparticle collision result in higher coagulation rates with accompanying higher settling velocities. This trend corroborates that found by Krone (1962), except that no "leveling off" of above a certain salinity value was found in these tests. The decrease in above a given salinity and concentration, as observed in both figures, usually represents the onset of hindered settling. The effect of salinity on is seen to be diminished at suspension concentrations in the hindered settling range. Owen (1971) found a negligible effect of salinity on the settling velocity of natural aggregates at two different locations in the Thames River estuary. The salinities at the two sampling stations varied between 6-10 ppt and 32-26 ppt, respectively. Evidently, the effect of salinity on Wj at these salinities in a turbulent flow field is much less than that under quiescent conditions (see Fig. 3.44). This implies 136 1.0 0.8 0.6 \ E 0.4 > 02 1- o o _l LU > Ql 0.08 _) t- 1- 0,06 LiJ CO 0.04 < Q LU 0.02 1 1 1 I — r"i I I I 1 — r-j-r Suspended Concentration o □ 0.25 g// I .0 g/-f 4.0 g/-? 16.0 g/^ 32.0 g/^ J — ^ L.J I M 2.0 40 6.0 8.0 10. 20. SALINITY, S (PPT) J 'III 40 60. 80. 100. Fig. 3.44. Effect of Salinity on Settling Velocity of Avonmouth Mud (after Owen, 1970). 138 that increased cohesion caused by the higher salinities is counter- balanced by the high internal shear rates which cause the aggregates to be broken apart. Deposition tests were conducted at the University of Florida in order to further investigate the effect on both salinity and bed shear stress on the settling rates of the Lake Francis mud. The salt concentrations utilized in these tests were: 0, 1, 2, 5, 10, 20 and 35 ppt. The tests were conducted in the rotating annular flume with a water depth of 0.31 m at the following values of Tj^: 0.0, 0.015, 0.10, 0.20 and 0.30 M/m . The initial concentration for these tests varied between 3,.7 and 4.7 g/1. Before the start of each test, the sediment and water were mixed for two hours at a shear stress of 0.90 N/m^. The shear stress was then reduced to the appropriate value and the samples of the suspended sediment were collected at 0, 1, 2, 5, 10, 15, 20, 30 and 60 minutes after the shear stress was reduced to t^^. Subsequent samples were collected with lower frequency. Each experiment was conducted for a period of 21 hours. The measured suspension concentrations were plotted against time for each experiment. Plots of C/Cq versus time for each value at a salt concentration of 5 ppt are shown in Fig. 3.46. The steady state concentration, Cg^, for each deposition test was determined from a curve drawn to represent the mean variation of the concentration with time, as seen in this figure. The following observations were made: 1) For the two lowest values of x^, i.e. = 0.0, and 0.015 M/m^, the concentration decreased over the duration of the experiment for all salt concentration values, indicating that C^^ for these x^ values would probably have been equal to zero if the experiment had been of longer 139 140 duration. 2) For = 0.05, 0.1. 0.2 and 0.3 N/m^ the concentration decreased rapidly during the first hour and C^q was reached more rapidly as the value of i^ increased. 3) At the four lowest values of t;|^ the effect of salt concentration on the deposition rate (i.e. concentration variation with time) was appreciable. For the two highest values the salt effect was much less discernable. These results seem to indicate that at relatively low values of cohesive forces are predominant, whereas at the higher values the hydrodynamic forces (i.e. disruptive internal shears) become at least as significant. This explanation follows from the results obtained by Owen (1970; 1971) and by Mehta and Partheniades (1975). The ratio Cgq/C^ was plotted against values for all salt concentrations (Fig. 3.47). Interpolation of the resulting plot yields ' ^""^ believed that even though additional data (i.e. Cgq/Cg against i^ values) might have resulted in a different value of "^b . ' values would be reasonable close (probably within ± mi n r J 25%). Analysis of the results from these experiments is given in the next section. 3.6.3. Deposition Rates The product P^.W^ in Eq. 3.28 defines an effective settling velocity, W^, which, in general, is smaller in magnitude than since the range of P^ is between 0 and 1. The rate of deposition given by Eq. 3.28 may therefore be written as dC s — = (3.41) dt d 141 Range I C Hindered Settling j RangelB I Mutual I Interference Range lA No Mutual Interference .Not a straigtit line in general because T^^ = f (C) Range E B Mutual Interference Range H A No Mutuallnterference 0.25 1.0 •-I b b bmin b max 3.48. Apparent Settling Velocity Description in Domains Defined by Suspended Sediment Concentration and Bed Shear Stress. 142 where is hereafter defined as the effective mean settling velocity for a given sediment. For the dimensionless bed shear stress less * * * than a certain characteristic value, Xj^ ^, with the range 0 ^\<'^^ designated as Range I, and for the concentration range C<C-^ for all values of (Fig. 3.48) the following empirical relationships for are assumed: Pd^,l for C < C 1 (3.42) s ,1 P .KC d for < C < (3.43) ( ) w (_ _ for c > C, 250 (3.44) IBv where W^j = median sediment settling velocity in the free settling range, gD^(P3/p^-l)/(18v) = and is defined by Eq. 3.29. Therefore, depending upon the value of C, the rate of deposition in Range I (see Fig. 3.48) is given by Eqs. 3.42-3.44. These three expressions for W^j are based upon the experimental results of Krone (1962), Ov^en (1971) and Bellessort (1973). Typical values for and €3 are 0.1-0„7 g/1 and 10-15 g/1 respectively. The value of 1 was found to be approximately -0.6 using the settling velocities measured by Owen (1970), Huang etal_. (1980) and Thorn (1981). The values W. , K, n and s , i Ci can vary widely, depending upon, among other factors, the particle diameter, D, the type of sediment and the salinity. These parameters must be determined in laboratory settling tests (further discussion of this aspect is given in Appendix D, Section D.2). 143 ^max ~ ^max Kin> ^""^ concentration range C > C^, designated as Range IIB, the rate of deposition is determined using a log-normal relationship (Mehta and Partheniades, 1975): dC -0.434 exp(-T^/2) dt 2/2^0^ t ^° * 2.04 ^V^^ (l-erf( loginf^- D) (3.45) /2 ^«>^P(-l-27^bmin) where = 0.49. Eqs. 3.32-3.34 and C* = iC^-Cf iC^-C^^) have been substituted into Eq. 3.37. The following expression for in Range IIB was determined by equating Eqs. 3.41 and 3.45: 0 k^d exp(-T^/2) C = o^t C 2.04 (V^^ (l-erf(_log^ [ ])) (3.46) /2 4exp(-1.27x. . ) bmin * where l<2 = 0.434/ (2/2Tt). For x^<l, the argument of the error function is set equal to zero. This expression for {and therefore Eq. 3.45 as well) is assumed to be valid for C>Ci for the following reason: the phenomenon of hindered settling was not observed in the steady-state deposition tests under turbulent flows performed by Mehta (1973) for concentrations up to about 20 g/1 . Evidently, the higher t* values that Mehta used in his tests prevented the occurence of this mode of settling, inasmuch as Krone (1962) did observe hindered settling in his tests which, in general, were conducted at lower values of -cb- 144 * Deposition tests with 0.25 < t:^ < 1 using San Francisco Bay mud in sea water and kaolinite in distilled water revealed that for suspension concentrations less than Cj^ ==0.1-0. 7 g/1 , the exponential law given by Eq. 3.41 was valid. Therefore, for C < C^^ in Range IIA, the rate of deposition is given by Eqs. 3.41 and 3.42 with obtained from: p = for T < 1 < x (3.47) d ,, b,c b b VJ 1 max s,l where W^^j is given by Eq. 3.46 with Cg=C=Cj|^. Thus, for C<C-|^ in Range II is defined such that the value of ii(Cj) in Range II is equal to i=Pcl*^''s,l C<C-|^. Therefore, and dC/dt are continuous functions for all concentrations in Range II. Likewise, the parameter t:^ ^ is defined to be the value of -t:^ at which the expression for ^^(C) in Range I is equal to the same in Range II. Thus, Wg and therefore dC/dt are continuous functions for the * ★ entire deposition range (t;^ < 1). It is apparent that Tu is not a max constant, as it is a function of the depth-averaged concentration, C. * Solving for ^ gives ^s II -^b,c = l-^- forC>C^ (3.48) * where Wgjj is given by Eq. 3.46, and W^^j by either Eq. 3.43 or 3.44 (depending on the value of C) divided by P^. 145 The previously described deposition tests performed in the annular flume using Lake Francis sediment in water with varying salinity were analyzed in order to determine the combined effect of salinity and bed shear stress on the settling rates of this sediment. The analysis performed is described next. The settling velocity at the time when 50% of the depositable sediment had deposited, i.e. tsg. was determined for each experiment in Range I in order to quantitatively evaluate the effect of salt concentration, and possibly the bed shear stress, on the rate of deposition. This particular value of the settling velocity, designated as WggQ, was chosen for this analysis because it can be shown to be more representative of the deposition rates in the time interval of interest in numerical simulation (see Section 3.6.4. for discussion of this aspect) than either the mean or the median value. The analysis was performed in the following manner for the experiments in Range I (i.e. 1^ = 0.0 and 0.15 N/m''). Equation 3.28 was integrated and rearranged to yield the given expression for W^jq: C -C -d ^0 ^50 W = An ( ) (3.49) Pd^50 ^0%q where C^q = suspension concentration at time t^Q. Values of W^^q computed from Eq. 3.49 are shown in Fig. 3.49 superposed on Fig. 3.44, with lines of equal shear stress drawn as indicated. The average value of the initial concentration, Cq. in these experiments was 4.2 g/1 . No consistent trend between and W rg (e.g. increasing values with 146 1.0 0.8 _ 0.6 1/1 1 — i — 111' E E >- o o _) UJ > UJ C/5 0.4 0.2' 0. 0.08- Q06 004 0.02- TTT (N/m^) • 0.0 * 0.015 Average Values Owen s Data Suspended Concentration o A □ 0.25 1 .0 4.0 16.0 32.0 g/f g/-^ J L I I I J I I I M .0 20 4,0 6.0 8.0 10. 20. SALINITY. S (PPT) 40. 60 80.100. Fig. 3.49. Effect of Salinity and Bed Shear Stress on Settling Velocity of Lake Francis Sediment. 147 increasing values for all salt concentrations) is apparent in Fig. 3.49. This observation suggests that W^gQ may be considered to be invariant with respect to i^i^ in Range I. Due to the limited data obtained, as well as the noted invariance of with respect to t:^ for Range I, the values of W^gg for the two values were averaged for each salt concentration. These average values, "Ws50» ^^''^ plotted against salt concentration in Fig. 3.49. Such an averaging procedure was performed in order to further investigate the effect of salt concentration on W^gg. A power curve relationship between Vl^gg and the salinity, S, of the following form was desired: WgggCS.C) = A W^5q(35,C) (3.50) where ¥55g(35,C) = KC" (K and n are defined in Section 3.6.1.) and A and m are empirical constants. It was proposed that when S was less than 0.1 ppt, its value would be set equal to 0.1, so that W55g(S<0.1,C) would be greater than zero and in fact W55g(S<0.1,C) = Wg5g(S=0.1,C) . The value of V^gg at S = 35 ppt was utilized as the W55g(35,C) value. Least squares linear regression analysis was used to determine if the averaged settling velocities followed such a power curve. This analysis gave the following values for A and m and the coefficient of determination, r^: A = 0.57, m = 0.13 and r^ = 0.96. As indicated by this r^' value, a good agreement was obtained between Eq. 3.50 and the data. This confirms that, at least for these experiments and the analysis method employed, the effect of salt concentration on W^gg in Range I can be expressed by a power relationship of the form given in Eq. 3.50. 148 The function given in Eq. 3.50 is incorporated into the deposition rate expression for Range I and Range IIB as follows: Equation 3.50 is used to evaluate the settling velocity as a function of the concentration of dissolved salt and suspended sediment. Based on the variety of relationships found between W^^g and C for several of these experiments, the following general expressions for W^gQ in Range I and for C < in Range II have been incorporated into the deposition algorithm: m W m = A,.W .S for C < (3.51a) SoU 1 I n, m^ ^s50 " '^l*^ ^1 ^ *^ ^ ^2 (3.51b) I n^ m. ^s50 " ^2 < C < (3.51c) 2 ^ A-gD (- — 1) p 250 3"^ P C n m- W = ( ■) ( 1) -S for C > C- (3.51d) pl.8 18V C3 where K^- = A^-K^- for i=l,2, and W^,^ is the constant settling velocity for C < Cj^. For concentrations greater than C^^ and less than C3, where hindered settling begins, two different expressions of the form W^i^q = K'C"s'" can be used to express the variation of W^^q with C and S. Two relationships were revealed in this concentration range in several of 149 the deposition tests (e.g. see Fig. 3.50). These relationships are permissible in the deposition/settling algorithm. If only one relationship is revealed from the log-log plot of W^^q against C, C3 must be set equal to The values of ^^ and m^ apply for C < C2 while A2 and m^ are for C2 ^ C < C3, and A3 and m3 apply for C > C3; if C3 = C2, then A2 = ^l and m2 = m^. The values of K^^ and n^ apply for < C < C2, while K2 and n2 are for < C < C3; if C3 = C2, then K^,^ = K2 and n-^ = n2. The values of K3 and n3 apply for C > C3. For the deposition-salinity experiments in Range II (i.e. = 0.05, 0.1, 0.2, and 0.3 N/m ) the following analysis was performed. * Figure 3.51 shows the relationship between C^q and the salt concentration for the values \ > \ . where Cg > 0. Based on the mi n nature of the equal curves in this figure and taking into consideration the limited number of deposition tests performed at > * Tu , C is assumed to be invariant with respect to salinity, '-'min In Range IIB, the effect of salt concentration on the deposition rate was evaluated in the following manner. The value of dC/dt at t=t5g was determined for each experiment. Substituting t=t5Q into eq. 3.37 gives the following expression for the rate of deposition at tgg: dC -0-^3^ Co(l<q^ -'50= (3.52) /2^ a t 2 ^50 The value of t5g for each experiment was determined from the log-normal * plot of C versus t, and the value of 02 was evaluated according to (Aitchison and Brown, 1957): 150 £ £ Q3 0.2 _ 0.10 0.08 >: 0.06 5 0.05 3 004 > 0.03- o □ 0.02 H- H LU CO 0.01 0 ^0.0 N/m S -O.ppt 1 — r 2 "1 — r 1 III 02 0.4 0.6 0.8 1.0 2.0 CONCENTRATION, Cig/i) 4.0 Fig. 3.50. Settling Velocity Versus Suspension Concentration for Deposition Test with Lake Francis Sediment. 0.6 =0.3 N/m 0.2 N/rrf 0.1 N/m^ 0.05 N/m' 10 15 20 25 30 SALINITY. S { PPT) 35 Fig. 3.51. Variation of C^^ with Salinity and x^. 151 (3.53) * where tj^g and tg^ are the times at which C = 16% and 84%, respectively, and likewise were determined from the log-normal plot. Due to the limited data and observed invariance of (dC/dt)|5Q with respect to t^j, the values of (dC/dt)|50 were averaged for each salt concentration. The following functional relationship between T (given by Eq. 3.36) and the salt concentration, S, was obtained using a linear regression analysis: where b = 10.78. f = -0.33 and = 0.93. When S < 0.1 ppt, S is set equal to 0.1 ppt. The effect of salinity on the deposition rate expression for Range TIB was incorporated by substituting Eq. 3.54 into Eq. 3.37. The resulting value of the time-rate of change of concentration, dC/dt, will decrease monotonically at any given time with decreasing salinity, while at the same time, the expression for C given by Eq. 3.35 still approaches 0 as t-^. This methodology of incorporating the effect of T = Tog^QCf )BS^] ^50 (3.54) * salinity requires that the values of tgg and cr^, used in Eq. 3.37, be evaluated at a salinity of 35 ppt. 152 3.6.4. Deposition Algorithm Deposition of suspended cohesive sediment occurs when 1) the flow is decelerating, i.e. Xj^(t+At) < ^^^(t), and 2) when tj^(t+At) < '^b^^^- VJhen these two conditions are satisfied at any node, the rate of deposition is calculated as follows. The value of t^jj^^ is evaluated using Eq. 3.48. Inasmuch as the log-normal relationship was found not to be suitable for \ < 0.25, the minimum x^^^ value is set equal to 0.25. Therefore, Eqs. 3.28, 3.29 and 3.51a are always used to calculate * * the value of dC/dt for 0.0 ^ \ \ c (^^"9^ ^h^^Q Eqs. 3.28, 3.47 and 3.51a are used for Range IIA. The maximum allowable value for i^j^^^, is 1.00; therefore, Eq. 3.45 is used to determine dC/dt in Range IIB. The amount of dry mass of sediment deposited per time-step element, Mq, is determined according to dC Mr, = ~ At d„ (3.55) ° dt ^ where d_ = V2 (d (t)+d (t+At) ) and dC/dt is given by 1 dC dC -[— +— ] (3.56) 2 dt^ dt^^^^ in Range I and Range IIA, and 153 C * At 0.434 f3 — (1 - C) [erf(Jln( ) + 1] in Range IIB. In Eq. 3.57. e = \[Q{t) + e(t+At)), where 9 = C, tgQ, Cgq and o^. Equation 3.57 was obtained by integration of Eq. 3.45 from t=0 to t=At. The thickness of the bed formed by Mq is calculated in the bed formation algorithm using the procedure described in Section 3.3.2. The sink term in the governing equation (Eq. 3.5) is given by Eq. 3.56 in Range I and Range IIA, and by Eq. 3.57 for Range IIB. For unsteady flows, as occur in estuaries, the value of C^^, which is the steady state value of the suspended sediment concentration found in laboratory tests under steady flows, is assumed to be zero. Nevertheless, the laboratory determined log-normal relationship for dC/dt, as given by Eq. 3.57, is used for Range IIB in the deposition algorithm for the following reasons. The time-step. At, used in the estuarial sediment transport problems is typically of the order 0.1t5o<At<10t5g. Therefore, dC/dt (given by Eq. 3.57) is significantly greater than zero at time At. This implies that after any time interval At, the suspended sediment concentration, C(t+At), does not approach Cp_, but is assumed to be equal to the following: dC C(t+At) = C(t) At (3.58) dt where dC/dt is given by Eq. 3.57. Thus, for unsteady flow conditions, the rate of deposition is considered to be a function of Cgq (given by Eqs. 3.32-3.34). Such a consideration is required for a realistic 154 interpretation of laboratory deposition test results for the purpose of ascertaining the depositional rates in the unsteady estuarial environment. 3.7. Consolidation 3.7.1. Description In this section consolidation of a freshly deposited cohesive sediment bed is described, followed by a review of some of the research that has been conducted on the consolidation of sedimenting clays, and a discussion on the possible correlation between the density and shear strength of such soils. As described previously, an estuarial sediment bed is formed when deposited sediment particles and/or aggregates comprising a stationary suspension begin to interact and form a soil which transmits an effective stress by virtue of particle to particle contacts. The self- weight of the particles, as well as the deposition of additional material brings the particles closer together by the expulsion of pore water between the particles and/or aggregates. Thus, consolidation is caused by the self-weight of the sediment particles (Parker and Lee, 1979). According to Parker and Lee, a soil is formed when the water content of the sediment-water suspension decreases to the fluid limit. Unfortunately, there is not a unique water content value for cohesive soils at which the suspension changes into soil (Been and Sills, 1981). This critical water content (i.e. fluid limit) is a function of the initial water content of the suspension. Prior to soil formation, i.e. for water contents above the fluid limit, the suspension is in a stress-free state. 155 During the transition from suspension to soil, an extremely compressible soil framework or skeleton develops (Been and Sills, 1981). The strains involved in this first stage of consolidation are relatively large. For example, Parker and Lee (1979) assumed that strain is one-dimensional and found that "the strain involved in the consolidation of an element of suspension, from say 10 g/1 (assume particle density 2650 kg/m-^, sea water density 1030 kg/m"^) to the commonly observed density of 1200 kg/ni^ would be derived from an initial solids fraction of 0.0038 and a final solids fraction of 0.1064 and show a 96% strain in the element." Lee aiid Sills (1979) state that this consolidation strain may continue for several days, or even months. The straining and upward expulsion of pore water gradually decreases as the soil skeleton continues to develop. Eventually this skeleton reaches a state of equilibrium with the normal stress of the overlying sediment (Parker and Lee, 1979). During the early stages of consolidation the self-weight of the soil mass near the bed surface is balanced by the seepage force induced by the upward flow of pore water from the underlying sediment. As a result, the effective stresses acting in the near surface region are very small and in general are not measurable (Been and Sills, 1981). As the soil continues to undergo self-weight consolidation and the upward flux of pore water lessens, the self-weight of this near surface soil gradually turns into an effective stress. This surface stress, and indeed the stress throughout the soil may first crush the aggregate structure of the soil and then crush the floes themselves. Primary consolidation is defined to end when the excessive pore water pressure has completely dissipated (Spangler and Handy, 1982). 156 Secondary consolidation, which may continue for many weeks or months, is the result of plastic deformation of the soil under a constant overburden. Figure 3.52 shows the variation of the mean dry bed density, p, with consolidation time for Avonmouth mud (Owen, 1977), commercial grade kaolinite in salt water (S=35 ppt) (Parchure, 1980) and for kaolinite in tap water (S=0 ppt) (Dixit, 1982). Noteworthy is the very rapid increase in p in approximately the first 48 hours, after which the increase was much less rapid, and the almost asymtopic approach to the final mean bed density, p^. Figure 3.53 shows the variation of the normalized mean bed density with consolidation time for the same three mud beds. This figure shows that, at least for these three mud beds, the time-variation of the degree of bed consolidation with time was approximately equal. Figure 3.54 shows the dimensionless relationship found between the bed bulk density, Pg, and the depth below the bed surface, z^, with consolidation times of the order of 48 hours for four natural muds. Figures 3.55a and 3.55b show the dimensionless relationship found by Dixit (1982) for T^^^ up to 24 hours and greater than 48 hours, respectively, for kaolinite beds in tap water. Figures 3.56-3.58 show the density profiles in Figs. 3.2-3.4 replotted on log-log paper. A brief review of the studies on the consolidation of clays is given below. Terzaghi (1924) developed the first theory governing one- dimensional primary consolidation in soils. This theory was based on several assumptions: 1) homogeneous soil, 2) one-dimensional compression, 3) one-dimensional (vertical) flow, 4) the self-weight of 157 TIME (hrs) Fig. 3.52, Variation of Mean Bed Density with Consolidation Time (after Dixit, 1982). ■ 0 40 80 120 160 TIME (hrs) Fig. 3.53. Variation of p/p with Consolidation Time (after Dixit, 1982). 158 3.54. z^/H Versus p/p for Avonmouth, Brisbane, Grangemouth and Belawan Muds (after Dixit, 1982). 159 Fig. 3.55. z^/H Versus p/p for Consolidation Times (a) Less Than 48 Hours and (b) Greater Than 48 Hours (after Dixit, 1982). 160 X 1 1.0 0.5 0.2 ^ 4.17 o 8.33 o\ \ 1 1 ^ \ 1 0.3 0.5 1.0 p/p 2.0 3.0 Fig. 3.56. Normalized Bed Density Profiles for Thames Mud for Two Different Consolidation Times. X 1 X 1.0 0.5 0.2 Ql 005 1 1 <\ \Q ^\ Tjj(-=2.78 days S(ppt) . 0 0 2 X 4 <^ 8 ' 16 0 32 i 1 1 02 0.5 1.0 p/p 2.0 5.0 Fig. 3.57. Normalized Bed Density Profiles for Avonmouth Mud as a Function of Salinity. 161 1.0 0.50 020 0.1 s ^ 0.05- 0.02- 0.01 0.2 278 Days 17.3 ppt 16.7 g/-^ ■0 ■■ P ^ (m) (m) (kq/m^) . 2.0 0.13 262 o 4.5 0.32 251 ^ 70 0.58 211 a 10.0 075 194 0.5 2.0 3.0 50 P / P Fig. 3.58. Normalized Bed Density Profiles for Different Bed Thicknesses . 162 the soil particles is neglected and 5) there is a constant relationship between void ratio and permeability and between void ratio and effective stress. The fifth assumption implies that the strain is assumed to be infinitesimal with respect to the thickness of the soil layer. All these assumptions limit the applicability of this theory to relatively stiff thin layers at large depths. Therefore, it can not be used in calculating the consolidation of soft sediment deposits, such as occur in estuaries, in which large strains are not uncommon and the relationship between void ratio and effective stress is not a constant (Einsele et_al_., 1974). Gibson _et_^. (1967) developed the first completely general theory of one-dimensional primary consolidation of soils. The assumptions inherent in this theory are: 1) the soil is saturated and consists of an incompressible pore fluid and a compressible soil framework, 2) the pore fluid velocities are governed by Darcy's law, and 3) there is a unique relationship between the void ratio and soil permeability and between the void ratio and effective stress. The governing equation for this finite strain theory is t d k(e) ae 1 5 k{e) da de 1 Qe _ [ .] + [ ] + = 0 (3.59) de 1+e az s-1 az (1+e) de az S-1 at w where e = void ratio, Z = time independent material coordinate, which is a measure of the volume of solid particles only, k = soil permeability, S = soil specific weight, ^ = unit weight of the pore fluid and a' _ effective stress. This theory can be applied to thick soft sediment 163 deposits because it includes the effects of the self-weight of the sediment particles, is independent of the degree of strain and includes the constitutive relationships between e and k and between e and o (Cargill, 1982). Because of the non-linearity of Eq. 3.59, general closed form solutions have not yet been obtained. Therefore, numerical methods must be used to solve this equation for e(Z,t) for given constitutive k and a relationships and boundary conditions. Finite strain consolidation computer programs which solve Eq. 3.59 using explicit finite difference techniques have been developed by Gibson et__a2_. (1981), Schiffman and Cargill (1981) and most recently by Cargill (1982). The required constitutive relationships may be determined using any one of the following devices: geotechnical centrifuge, stress-controlled slurry consolidometer, pore pressure probe and nuclear densitometer (Croce, 1982; Znidarcic, 1982). One important limitation of finite strain consolidation theory is that since the effective stress at the surface is assumed to be zero for all times, the void ratio there remains a constant. This differs, as mentioned previously, from the observed decrease in void ratio (which is directly related to p) at the surface of soft sediment beds with time (Lee and Sills, 1981). Lee and Sills (1981) assumed that the two constitutive relationships were linear, and that the consolidation coefficient dcJ P^(l+p) de (3.60) ! 164 1 was a constant and solved Eq. 3.59 analytically. They accounted for the j decrease 1n the void ratio at the soil surface with time by adding an imaginary overburden of specified thickness on top of the actual bed. Comparison of measured density profiles for a clayey silt with those predicted by the analytical model yielded satisfactory results (Been and Sills, 1981). However, inadequacies of the assumptions made by Lee and Sills (1981) are reflected in these comparisons. Next, a discussion is presented on the shear strength characteristics of cohesive soils and the possible correlation between the density and shear strength of these soils. The shear strength of clays are due to the frictional resistance and interlocking between particles (physical component), and interparticle forces (physicochemical component) (Karcz and Shanmugam, 1974; Parchure, 1980). Taylor (1948) states that some of the factors j which affect the shear strength of clays are: type of clay mineral, water content, consolidation tine, stress history, degree of sample disturbance, chemical bonding (cohesion), anisotropy and exchange of cations. Consolidation results in increasing bed density and shear strength (Hanzawa and Kishida, 1981). Figure 3.59 shows the increase in the shear strength profile with consolidation time for flow-deposited kaolinite beds in tap v/ater. As mentioned previously, the nature and effect of consolidation on the shear strength profiles of cohesive sediment beds are not well known at present, and what limited information has been obtained is often contradictory (Parchure, 1980). Several researchers have attempted to establish a correlation between the bed density and shear strength of clay beds in order to implicitly evaluate the effect of consolidation on i^c(Z). The results of these efforts are summarized below. 165 Fig. 3.59. Variation of (Z|^) with for Various Consolidation Periods (after Dixit, 1982), I i 166 Figure 3.60 shows the correlation Owen (1970) found between the dry sediment density and shear strength for statically deposited beds of Avonmouth mud. Least squares analysis of the data plotted in this figure gave a slope of 2.44 and a coefficient of determination of r^ = 0.83. The power expression relating i^ and p is of the form \ = "'P ^ (3.61) with a = 5.85x10"^ and P = 2.44. Owen considered that the correlation obtained between and p was satisfactory, considering the experimental error involved in the measurement of both these parameters. Thorn and Parsons (1980) likewise found a power relationship between and the dry sediment density at the bed surface, „ , for Grangemouth, Belawan and Brisbane muds in saline water. They obtained values of 5.42 x 10 and 2.28 for a and p, respectively. In an earlier study, however, these researchers observed no significant correlation between and p^ for Grangemouth mud (Thorn and Parsons, 1977), while Thorn (1981) obtained a linear relationship between i^ and p^ for mud from Scheldt, Belgium in 5 ppt water. Bain (1981) found a relationship of the following form between e and for Mersey and Grangemouth muds: ^ P (3.62) 167 10.0 X UJ m cr < LU X m 5.0 1.0 05 1 1 1 c 1 s 1 ! d 1 1 /I ■ (mg/ ) (m) ~ o 16.290 32.9 10.06 - • 15.520 17.0 9.38 ▼ 17.475 17.8 6.98 J3 / ^ / ■ 1 1 .dOKJ lb. ^ 4.04 7 9 6.705 2.7 9.73 7.288 4.6 9.74 / — O 8.392 8.8 a76 ^ a 10.272 16.8 1002 ^ X / 6.866 33.3 9.72 A 6 810 0 8 9 74 h ▼ ' •Onr/ ■ • T ^ / - • / X - ▼ ▼ p « a / / oo / ° h a / o ft X 1 1 / 1 1 1 1 1 1 60 80 100 150 250 DENSITY ^(kg/m^) 350 Fig. 3.60. Correlation of Bed Shear Strength with Bed Density (after Owen, 1970). 168 with C = 2 X 10^'^ N/m^ and ^ = -18.3 for f^ersey mud, and C = 8 x 10'' and ^ = -6.1 for Grangemouth mud. The diversity of these relationships serves to emphasize that such properties of cohesive sediment soils must be established for each sediment studied. A description of the consolidation algorithm is given in the next section. 3.7.2. Consolidation Algorithm Consolidation of the deposited sediment bed is accounted for by increasing the bed density and shear strength with time. These calculations are made on an element-by-element basis. The description given for the consolidation algorithm developed as part of this study is for any one given element. Consolidation is considered to begin after the bed formation process is complete, at which time the bed thickness will be maximum. As described in Section 3.3.2 and Appendix D, Section D.2, the discretized density profile at a consolidation time, T^^., of a certain magnitude T^^^^ (selected to be equal to two hours as measured in laboratory tests) is used in the bed formation routine to form the bed structure resulting from deposition of a given mass of sediment. The consolidation period for a bed begins the first time step during which no deposition is predicted to occur. After Tj^j hours of consolidation (i.e. 2 hrs.) the stationary suspension present on top of the bed becomes part of the partially consolidated bed, and therefore would undergo resuspension if subjected to an excess shear stress. The dry sediment mass of the stationary suspension is first determined, and then the bed formation routine is used to evaluate the thickness and structure of the partially 169 consolidated bed formed by this mass. This procedure accounts for the observation that after approximately two hours of consolidation a cohesive sediment bed undergoes resuspension when subjected to an excess shear, whereas for T^^^. < 2 hours the bed mass erodes when subjected to the same shear (Dixit, 1982). Should further deposition occur for T^^^ < 2 hours, the value of T^^. is reset to zero, and the new bed formed is evaluated using the crushing procedure in the bed formation routine. The increase in bed density and shear strength with time is simulated to begin at T^^^, = 2 hours. This is the reason why the measured bed density profile at T^^ = 2 hours is used in the bed formation algorithm. The procedure for evaluating p{z^,t) is given below. 1) First, the final mean bed density, is determined. ~„ and the time, Tj^^, at which p = Pa, have been shown to vary linearly with C^, which is defined to be the suspension concentration that would result if the entire partially consolidated bed was resuspended (Owen, 1970). is given by the following: pH (3.63) where p and H = mean bed density and bed thickness, respectively, of the partially consolidated bed at the end of the previous time step (i.e. t=t-At, where At = time step). The equations used for ~^ and T^ are 170 P = (p ) + aC °° 0 0 (3.64) T . = (T , ) + bC dc^ dc^'^ 0 (3.65) where (Poo)q and (T^^^ )q are the extrapolated (hypothetical) intercept values for = 0. These two parameters and a and b are empirical coefficients that must be determined experimentally. The value Pa, is defined to the value at t = T^^ at which the following criterion is satisfied: p(t) - P(t-At) < 10 -3 P(t) (3.66) with At taken to be 24 hours. 2) The value of p(t = T^^) is determined next. The following relationship for p is indicated by Fig. 3.53: P{T^c^/P»= 1 - f*exp(-p-T^,/T^^ ) (3.67) CO Least squares analysis of the three sets of data plotted in Fig. 3.53 gave f = 0.845, p = 6.576 and for the coefficient of determination r^ = 0.993. 3) The bed density profile P(z5,T^j.) is next determined. The density profiles given in Figs. 3.54-3.58 can be expressed as 171 H-z. = A{ )' (3.68) where the values of A and B are functions of time for T^^ < 48 hours, and constants for T^^ > 48 hours. This variation of the bed density profile with time is depicted in Fig. 3.61. The value of z' = [W-z^) IW below which A and B are invariant with respect to time is defined to be ^max* '^^^^ r^d^r\% that for T^^^ > 48 hours and for T^^, < 48 hours with z' ^ ^max' P''^b^ ^H- 3.68 with A and B equal to the respective values for T^^, > 48 hours, while for T^^ < 48 hours and z'^^^ < z' < 1.0, pfZfj) is given by Eq. 3.68 with the values of A and B being functions of time. The values of A, B and z^jj^^ determined for the density profiles in Fig. 3.55a are given in Table 3.3. Table 3.3 Variation of Empirical Coefficients the Relationship Between p{z^) and T^^^. Tj^( hours) 7' max 2 0.36 -1.40 0.43 5 0.48 -0.72 0.60 11 0.62 -0.45 0.76 24 0.66 -0.50 0.84 >48 0.80 -0.29 1.0 Fig. 3.61. Variation of p(z, ) with T,^ Incorporated in Consolidation Algorithm. ° 173 Least square analysis of the data given in the above table revealed that all three parameters. A, B and z^j^^^^, varied with J^^ according to ® " '^^dc'' "^dc ^ ^^^^^ ^^'^^^ with D = 0.32, -1.71 and 0.39, G = 0.24, -0.49 and 0.24, and r^ = 0.96, 0.95 and 0.96 for 9 = A, B and z,|,ax» '"'^specti vely. Figure 3.58 shows a bed density profile for which p was measured below z' = 0.05. Based on the best fit line drawn through the data points in this figure, it is assumed for 0.0 < z' < 0.05, p(z^) is equal to the constant value p(Z[^=0.95H). This extrapolation of the density profile down to the bottom of the bed, = H, is required for two reasons: 1) in order to determine p{Z|j=H), since the power law given by Eq. 3.68 cannot be evaluated at = H, and 2) in order to insure conservation of sediment mass in the bed. To summarize, p{z\^) is evaluated at a particular J^^ value as follows. For T^^ < 48 hours: '^"^^ n H-z, -1.71T, = 0.32T^^0-2^-^) dc (3^^^^ H 0 24 "-^b -1.71(48)-1-^^ = 0.32(48)"^^ -) (3.71) H 174 for 0.05 < z < z^^', max ^^^b^ 0.24 -1.71(48)"°*^^ = 0.32(48) (0.05) (3.72) = 1.75 for 0.0 < z < 0.05 , 0.24 where z^^^ = 0.39 T^^. . For T^^^. > 48 hours, Eqs. 3.71 and 3.72 are used with z'^^^ = 1.0. The discretized bed density profile used in the layered bed model is changed using Eqs. 3.70-3.72 to reflect the increase in bed density due to consolidation. 4) The thickness of the bed, H, is reduced to account for the expulsion of pore water during consolidation, and to insure that the mass of sediment in the bed is conserved. The rate of change of H with time is given by dH H dp dt=--dt (■^•73) P Using first order finite difference approximations for the two derivatives, Eq. 3.73 becomes p(t-At) - p(t) H(t) = H(t-At) {l+2( )} (3.74) p(t) + p(t-At) 175 where H(t) = bed thickness at the current time step and H(t-At) = bed thickness at the previous time step. The thickness of each bed layer is adjusted as follows: H{t)-H(t-At) T.(t) = T.(t-At){H } (3.75) ^ ^ H(t-At) where T^- = thickness of the i-l!l bed layer. 5) If further deposition occurs when T^^ > 2 hours, a new sediment bed is formed on top of the existing partially consolidated bed. To simulate the occurrence of such repeated periods of deposition, as typically occurs in estuaries due to the oscillating tidal flow, the new deposit portion of the bed model is further divided into a finite number of strata (Fig. 3.62). The top stratum may be composed of a stationary suspension and partially consolidated bed, whereas the buried strata are composed of just partially consolidated beds. The degree of consolidation of a particular stratum (in relation to that of the other strata) is accounted for by using a separate T^^ for each stratum. The bed density profile for the 1— stratum as a function of T^^.., where the sub-subscript i refers to the i-^ stratum, is determined as follows. Step 1 is performed with H = total bed thickness of all strata and 7 = mean bed density of all strata. Step 2 is repeated for each stratum. Thus a separate value of p is determined for each stratum. Steps 3 and 4 are likewise repeated for all the strata, with the total bed thickness used for H and a separate value of p used for each stratum. 176 Strata X ® iz X dc nr ir 'dc IE I ® Tdci Bed Surface Settled Bed z = I.O UNO PCND ^ PCND PCND PCND •z =0.0 "^dc "^dc ^ "^dc Ti-ip ULj ULjj UCjjj UCj2 Fig. 3.52. Bed Schemati zation Used in Bed Formation-Consolidation Algorithms . 177 Due to the extremely limited number of studies on the nature of shear strength profiles in cohesive sediment beds, the variation of the bed shear strength profile "^^^^b^ ^^'^'^ "'"dc ^'^ determined indirectly by use of a functional relationship between p and x^. The relationship found between these two parameters (Eq. 3.61) by Owen (1970) and Thorn and Parsons (1980) is used in the consolidation algorithm to account for the increase in with increasing T^^,. The empirical coefficients used in the consolidation algorithm (i.e. a, p, (p„)g, (T(J(;^)q, a, b, f, p. A, B, and z'^^^) must be determined by performing laboratory consolidation tests. A brief description of a test procedure is given in Appendix D, Section D.2. CHAPTER IV MODEL DEVELOPMENT 4.1. Introductory Note This chapter begins with a review and evaluation of previous cohesive sediment transport models, then follows with descriptions of the cohesive sediment transport model developed during this study, the finite element formulation used in the model, and lastly, the convergence and stability characteristics of the model. 4.2. Review of Previous Models One of the first cohesive sediment transport models was developed by Odd and Owen (1972). This was a two-layered, one-dimensional coupled model which simulated both the tidal flow and mud transport in a well- mixed estuary. The two layers were of unequal thickness, with uniform properties (e.g. flow velocity, suspension concentration) assumed for each layer. A rectangular flow cross-section was also assumed. The equations of motion and continuity for each layer were solved using a finite-difference formulation, while the advection-diffusion equations governing the transport of suspended sediments in two layers were solved using the method of characteristics. Erosion and deposition of sediment were simulated in this model. O'Connor and Zein (1974) developed an un-coupled two-dimensional, laterally averaged suspended sediment model which solves the advection- 178 179 diffusion equation using an implicit finite-difference method. Horizontal eddy diffusion is neglected, as is the assumed negligible vertical water motion in comparison with the sediment settling velocity. The settling and erosion of sediment was accounted for in some of the described model applications by modifying the vertical sediment diffusion coefficient. The model is strictly applicable to quasi-steady depth and flow conditions. It was later modified to include unsteady (tidal) flow conditions by O'Connor (1975). Ariathurai (1974) and Ariathurai and Krone (1976) developed an un- coupled two-dimensional, depth-averaged sediment transport model which uses the finite element method to solve the advection-dispersion equation. This model simulates the erosion, transport, aggregation and deposition of suspended cohesive sediments. Aggregation is accounted for by determining the sediment settling velocity as a function of the suspension concentration. Required data include the two-dimensional, depth-averaged velocity field, dispersion coefficients, and the sediment settling and erosion properties. Ariathurai et al_. (1977) modififed this model to solve the two-dimensional, laterally averaged suspended sediment transport problem. This latter model was verified using field observations in the Savannah River Estuary. Kuo^al_. (1978) developed a two-dimensional, laterally-averaged, coupled model which simulates the motion of water and suspended sediment in the turbidity maximum of an estuary. The vertical dimension is divided into a number of layers, and a finite difference method is used to solve the equations of motion, continuity and sediment mass balance for each layer. Erosion and deposition are accounted for in the mass balance equation for the bottom layer. 180 Koutitas and O'Connor (1980) developed a three-dimensional suspended sediment transport model which solves the advection-diffusion equation using a mixed finite difference-finite element method. Central finite differences are used in the horizontal directions while linear finite elements are used in the vertical direction. The two horizontal turbulent diffusion coefficients and the vertical water velocity are assumed negligible in comparison with vertical eddy diffusivity and sediment settling velocity, respectively. The source/sink term which accounts for the erosion and deposition of sediment is not included in the governing equation. Cole and Miles (1983) describe a two-dimensional, depth-averaged model of mud transport which solves the advection-dispersion transport equation by a finite difference method. Deposition and dispersive transport are simulated, but erosion is not. None of these fine sediment transport models consider the following two factors. 1) Consolidation of the mud bed and the effect this has on the erodibility when the bed is subjected to an excess bed shear stress. 2) The effect of salinity variation (e.g. in the mixing zone between fresh and sea water in estuaries) on the processes of erosion and deposition of cohesive sediments in a turbulent flow field, since the empirical laws used to determine the rates of erosion and deposition were derived using empirical evidence from laboratory experiments conducted in natural or artificial sea water. In addition, the empirical laws of erosion and deposition used in these models cannot be considered to be "the state-of-the-art" even for sea water, as a considerable number of laboratory tests conducted since these laws were proposed have revealed new evidence on the erosional and depositional 181 behavior of cohesive sediments. For example, the empirical erosion rate expression used in the existing models is given by Eq. 3.13, which, as described in Chapter III, has been found to be applicable only to settled beds and not to partially consolidated beds. Likewise, the empirical deposition rate expression used in these models (Eq. 3.50) is limited to only a small percentage (e.g. approximately 20% for kaolinite in tap water) of the bed shear stress range over which subsequent laboratory tests have shown that cohesive sediments deposit under steady flow conditions (Mehta, 1973). 4.3. Model Description The cohesive sediment transport model, referred to hereafter as CSTH-H, developed during this study is a time varying, two-dimensional, un-coupled finite element model that is capable of predicting the horizontal and temporal variations in the depth-averaged suspended concentrations of cohesive sediments and bed surface elevations in an estuary, coastal waterway or river. In addition, it can be used to predict the steady-state or unsteady transport of any conservative substance or non-conservative constituent, if the reaction rates are known. CSTM-H simulates the advective and dispersive transport of suspended or dissolved constituents, the aggregation, deposition and erosion of cohesive sediments to and from the bed, respectively, and the consolidation of the bed. CSTM-H is composed of the algorithms and layered bed model developed in Chapter III integrated into a modi_fied version of the finite element solution routine developed by Ariathurai (1974). A description of this finite element formulation and the modifications 182 made to it is given in the next section. A synopsis of the operations performed by CSTM-H during each time step is given below. The flow chart given in Appendix C, Section C.3 depicts the step-by-step solution procedure incorporated in CSTM-H. The average bed shear stress induced by the turbulent flow velocity of the suspending fluid is calculated for each element. Then the amount of sediment, if any, that was deposited onto or resuspended from the bed in each element during the previous time step is determined using the deposition and erosion algorithms, respectively. The dispersion algorithm then calculates the values of the four components of the two- dimensional sediment dispersivity tensor. Using these values and the prescribed velocity field and concentration boundary conditions, Eq. 3.5 is solved for the suspended sediment concentration at each node for the next time step. The new bed elevation in each element is determined by adding or subtracting the thickness of sediment deposited onto or resuspended from, respectively, the bed profile that existed during the previous time step. Lastly, the consolidation algorithm calculates for each element the increase in bed density and shear strength and the decrease in bed thickness due to consolidation during the previous time step. The following five types of data are required to apply CSTM-H to a particular water body: 1) input/output and transient control parameters, 2) finite element grid of the system to be modeled, 3) two- dimensional depth-averaged velocity and salinity fields, 4) concentration initial and boundary conditions and 5) properties of the cohesive sediments in the water body to be modeled which characterize their erosion, deposition and consolidation. The user's manual in 183 Appendix C, Section C.4 lists the required information in each of these five data groups. A brief description of the five groups is given below. The required input/output and transient control parameters include the time step size, the degree of implicitness used in solving the temporal problem, and code arrays which specify at which time steps new values of various parameters (e.g. depth of flow) are read in. These transient code arrays also specify the type of output (e.g. nodal concentrations and/or discretized bed profiles), if any, required at each time step. The finite element grid of the water body to be modeled is defined by the number of elements and nodes the water body is divided into, the two horizontal coordinates of each node point, and the number of nodes which form each element. Quadratic and/or triangular elements with curved sides may be used in CSTM-H. A finite element grid generating program is a helpful tool in generating and modifying a finite element grid. This is especially true when a relatively large body of water (e.g. Tampa Bay) is being modeled. The advantages of using the finite element method over the more conventional finite difference method in estuaries and other similar bodies of water are delineated in the next section. The velocity field is defined by the two horizontal components of the depth-averaged flow velocity and the depth of flow at each node and time step. The only practical methods available today to determine the velocity field in the detail required by an un-coupled transport model such as CSTM-H to model an estuary or other prototype water body are physical and mathematical models. The advantages and disadvantages of 184 each are well documented, and therefore will not be discussed here. Numerous two-dimensional hydrodynamic mathematical models have been developed; these are likewise well documented in engineering literature. The FESWMS hydrodynamic model developed by the Water Resources Division of the U.S. Geological Survey is particularly well suited for modeling the velocity and salinity fields because this model uses the same basic finite element formulation (i.e. isoparametric quadrilateral and/or triangular elements with parabolic sides) that is used in CSTM-H. Therefore, the same grid can be used in both models. The salinity (i.e. density) field needs to be determined for water bodies where spatial and/or temporal variations in the salinity occur, in order to model the effects that variations in the salinity have on the erosion and deposition characteristics of cohesive sediments. A description of the method used in CSTM-H to evaluate the bed shear stress over a cohesive sediment bed using the two depth-averaged horizontal velocity components is given later in this section. The depth-averaged suspended sediment concentration must be specified at each node at the start of the modeling effort (initial conditions). Boundary conditions (i.e. depth-averaged suspension concentrations or normal concentration flux) are required for all external v/ater boundaries of the system being modeled. For the nodes which define such external water boundaries at which no concentration boundary conditions are given, CSTM-H assumes that the spatial concentration gradient is zero. The boundary conditions at the free water surface and the bottom are expressed by Eqs. A. 35 and A. 36, respectively. Equation A. 35 expresses that there can be no net rate of sediment transport across the free water surface. The bottom boundary 185 condition (Eq. A. 36) expresses that eroded sediment material is transported in the vertical direction away from the bed by turbulent diffusion and that deposited sediment becomes part of the bed. Equations A. 35 and A. 36 are actually included in the governing equation (Eq. 3.5) since it is vertically integrated from the bottom to the free water surface (see Appendix A). The sediment parameters which prescribe the erosion, deposition, bed formation and consolidation characteristics of the sediment in the water body to be modeled are described in Chapter III, Sections 3.3, 3.4, 3.6 and 3.7 and in Appendix C, Section C.4. The field data collection and laboratory sediment testing programs recommended for obtaining the data required by CSTM-H are described in Appendix D. Descriptions of some additional functions incorporated in CSTM-H are given below. The settling velocity in Range IC is a function of the kinematic viscosity, v (see Eq. 3.59d). The following equation for v as a function of the mean water temperature, (which is read in the fifth data set above) was determined with data obtained from Bolz and Tuve (1976), using least squares linear regression analysis: V = 1.7017x10"^ • exp(-0.0251T ) (4.1) w where T^ is the mean water temperature in degrees Celsius and v has units of m /s. The coefficient of determination for this equation is 2 r =0.994, which indicates a good agreement between Eq. 4.1 and the data. The density of the suspending fluid (in Kg/rn^) is calculated at each node as a function of T^ and nodal salinity value using the following empirical equation (Wilson and Bradley, 1968): 186 P = 1000. 0*(0. 702 + 100. 0*(17. 5273 + O.llOlT - w w 0.000639T^ - 0.039986S -0.000107T S) w w (5881.913 + 37.592T - 0.34395T ^ + w w 2.2524S)"^) (4.2) where S = salinity in ppt. Incorporating, in addition, the effect of the suspended sediment on the local water density gives (MacArthur, 1979) e,w w s w '^s where Pg = effective local water density = f(T^,S,C), and p^^ is given by Eq. 4.2. At each time step where nodal salinity values change, new nodal values of p^ and Pg ^ are calculated. At each time step where nodal concentrations change, new nodal values of pg ^ are determined. The values of pg ^ are used in CSTM-H to calculate the nodal values of the bed shear stress tj^, while the values of p^ are used in determining the nodal values of the dry density, p, which are used in the erosion, bed formation and consolidation algorithms. The bed shear stress is calculated at every node using the following relationship between and u^ = friction velocity: \ = Pe,w"f^ (4.4) 187 Modal values for are determined as follows. The magnitude of the depth-averaged velocity vector, U, at each nodal point is given by - 2 2^/2 U = (u^ + v^) (4.5) where u and v are the two depth-averaged velocity components given as input data. The vertical velocity profile for a fully developed two- dimensional, bounded shear flow over a cohesive sediment bed, which Mehta (1973) and Gust (1976) have found to be hydrodynamically smooth, is given by (Chri stensen, 1977) U(2) •^f^ = 5.5 + 2.5 In ( 5.29) (4.6) where U(z) is the horizontal velocity component at an elevation z above the bottom. Integrating Eq. 4.5 over the local depth of flow, d, gives U — = 2.5 In (- 3.32 u^d - 17.56) (4.7) Using the value of U determined by Eq. 4.5, a Newton-Raphson iteration scheme is used to iterate for the value of u^. Then tj^ is determined using Eq. 4.4. A description of the FORTRAN computer program of CSTM-H is given in Appendix C. In Sections C.l and C.2, the functions of the main program 188 and subroutines are respectively described. A flow chart and user's manual are given in Sections C.3 and C.4, respectively. In the next section, the finite element routine used is described. 4.4. Finite Element Formulation 4.4.1. Introductory Mote The finite element method has been used to solve the governing equation (Eq. 3.5). This method is a numerical analysis technique for obtaining approximate solutions of differential equations. The discretization procedures used reduce the equation to be solved to one with a finite number of dependent variables by dividing the continuous solution domain into a number of elements and by expressing the dependent variable in terms of approximating interpolation (i.e. shape) functions within each element. The values of the dependent variable at node points are used to define the interpolation functions. Node points are usually located on the boundaries of elements and are used to define the connection between adjacent elements. The number and location of the node points must be chosen such that continuity of the dependent variable across common boundaries of adjacent elements is achieved (Zienkiewicz, 1977). The behavior of the dependent variable within each element is defined by the values of the dependent variable at the nodes and the shape function. Then the error which results from the use of the approximate dependent variable at each node in Eq. 3.5 is minimized,, This procedure results in a set of simultaneous equations which are solved for the unknown nodal dependent variables at the next time step,, A detailed description of the method is presented by Zienkiewicz (1977). 189 This method is preferred over the finite difference method because derivative boundary conditions do not require special treatment in the finite element method as they do in the former. It is a particularly advantageous method to use in estuarial transport problems because of the ability to use arbitrarily shaped elements. Quadrilateral and/or triangular elements may be used in CSTM-H in which a quadratic function approximation is used to describe both the intra-element spatial variation of the geometry and suspended sediment concentration. Therefore, the elements are isoparametric and may have curved sides. 4.4.2. Shape Functions The global and local element coordinate systems are shown in Figs. 4.1a and 4.1b, respectively. The global x-y coordinate system is continuous over the entire solution domain, while the local element ?,ri coordinate system applies only within an element. The local coordinate systems for a quadratic quadrilateral and quadratic right triangular element are shown in Fig. 4.1b. The local or area coordinates for the corner nodes 1, 3 and 5 of the triangular element are (0,0), (1,0) and (0,1), while those for the corner nodes 1, 3, 5 and 7 of the quadrilateral element are (-1,-1), (1,-1), (1,1) and (-1,1). Because three nodes are used along each edge of the triangular and quadrilateral elements, quadratic shape functions are required. The quadratic shape functions in CSTM-H determine the values of both the dependent variable, C, and the element geometry. Thus, the elements are isoparametric (Zienkiewicz, 1977). There is one shape function, N^-, for every node in a given element. Thus, for triangular elements there are 190 7 6 5 I 2 3 (a) Global Coordinates 5(0.1) i 2 (b) Local Coordinates Fig. 4.1, Global and Local Coordinates 191 six shape functions while for quadrilateral elements there are eight. The shape functions are functions of the local coordinates C and ti and the values of C and ri at the nodal points. The quadratic shape functions for quadrilateral and triangular elements are given in Table 4.1. The parameters C^- and ti^. in this table are the nodal coordinates. For example, for a quadrilateral element, r\. = -i, -i for node 1, while for a triangular element, 5^-, r\. = 1, 0 for node 3. The dependent variable, C, is approximated as the following function of the unknown nodal point concentrations, C^- , and the shape functions p N-: i=n C. = 2 N.C. J i=l ' ' (4.8) where Cj = approximate suspended sediment concentration at any location inside the jth element, and n = number of nodes forming the jth element. Likewise, the global coordinates x and y are approximated as the following functions of the global nodal point coordinates, x^- and y^-, and the shape functions, N-: i=n X = 2 N.X. i = l ^ ^ (4.9) 192 Table 4.1 Quadratic Shape Functions Quadrilateral Element Shape Function Node Number ]. = (l+^C.)(l+TlTl.)(^|.+TlTl.-l)/4 Corner Nodes 1,3,5,7 M^. = (l-^^)(l+TiTi^. )/2 Midsection Nodes 2,6 N^. = (1-Ti^)(l+?C. )/2 Midsection Nodes 4,8 Triangular Element Shape Function Node Number N. = 4^.1,, (1-Ti^. )(l-€^.-Ti^. ) Midsection Node 2 N^. = 4 l.r]./{l.+r].) Midsection Node 4 = 4 f]^{l-l.-r].){l-l.) Midsection Node 6 N. = U-l^--n.){l-2{l.+r\.-2l.r\.)} Corner Node 1 N. = -2lA/2-2-n:.-l.+r].^+'(].l.+r]./{l.+r].)} Corner Node 3 2 ^1 = ^T^l. -Ti . +1. +C. Ti. +C. / ( 5. +n. ) } Corner Node 5 193 The shape functions are used for two additional purposes: 1) to transform from the global coordinate system to the local element coordinate system and 2) to transform the derivatives of C with respect to X and y to the local element coordinates. To perform these transformations, the derivatives of with respect to x and y are needed. These are derived below. The derivatives of N^- with respect to x and y are given in terms of the derivatives of N^- with respect to the local coordinates C and t) using the chain rule of partial differentiation (Zienkiewicz, 1977) aN. ay aN. ay aw. ax ari a^ a^ a-n = [— • • — 1 ' iJf^ (4.10) aN^. ax aN^. ax m. — = C— • • — } ' iJi'^ (4.11) ay ac at] a-n a^ where |Jl is the Jacobian given by 1J| = ax ay a^ ac ax A ay ari (4.12) The derivatives of N^. with respect to I and ri for both quadrilateral and triangular elements are given in Table 4.2. 194 Table 4.2 Derivative of Shape Functions Derivative Quadrilateral Element Node Number ^^.(l+TlTi^.)(2^S^.+r)ri^.)/4 Corner Nodes 1,3,5,7 Midsection Nodes 2,6 ^i,.(l-ri^)/2 1 Midsection Nodes 4,8 n^.(l+CC.)(2TiTi.+^^.)/4 Corner Nodes 1,3,5,7 ^. (l-r)/2 Midsection Nodes 2,6 Derivative Midsection Nodes 4,8 Triangular Element (6 = Kronecker Delta) Node Number 4[l-2(C+Ti.)+2^.Ti+n.^]6.^. 4Ti.[lH./(C.+Ti.)]/(^.+n.)6.^. -4Ti.(2-2C.-..)6.. ( -3+45 . +8T1 . -81. ri. -471^ ) 6. . {N„ / ^ -2C . [ - 1+Ti . +n / ( 5, +ri . ) 2 ] } 6 -2ri.[-2+2l.+Ti.+l/(6.+n.)+C./(5.+Ti.)^]6.^. -45.(2-5.-2..)6.. 4?^.[l+n./(c.+n.)]/(5.4.n.)6.^. 4(l-?.)(l-5.-2Ti.)6.^. {-3+4Ti.+8C.-8C.ri.-4C^?)6_ -2C^ [-2+2T1 . +C . +1 / ( I, +n . ) +n . / { I. +T1 . ) ^ ] 6 . . {N5/T15-2T1. [-1+5. +C. / ( 5. +T1. ) 2] I. 2 4 6 1 3 5 2 4 6 1 3 5 195 The four components of the Jacobian given by Eq. 4.12 are equal to — = 2 X. — = Z X. i=i ac ^ an i=i an ^ (4.13) ay aM 5y . aN . i=n 1 i=n 1 ac 1=1 ac an i=i an Likewise, the derivatives of C with respect to C and n are equal to ac . „ aw. ac . an. i=n 1 i=n 1 — = 2 C. — = Z c, (4.14) ac i=i ac ^ an i=i an ^ 4.4.3. The Galerkin Weighted Residual Method The Galerkin weighted residual method has been used to solve the governing equation. This method requires that the summation of weighted residuals over the entire solution domain be equal to zero when the shape functions are used for the nodal weighting factors. The residual, r, results from applying the governing equation to the element subdomain using the approximate suspension concentration C instead of the actual concentration C. In order for C to satisfy all the stipulated boundary conditions, the sum of the normal concentration fluxes from adjacent elements and any source or sink must be equal to zero on all internal and external boundaries in the solution domain. This condition may be expressed mathematically as (Ariathurai and Krone, 1976): q^- + q^- + q^- 196 = 0 i=l....,NL (4.15) where q| = outward normal flux from one element = inward normal flux from adjacent element qf = normal flux from source/sink on the ith boundary NL = number of element interfaces and external boundaries The formulation of the Galerkin method can be expressed mathematically as j=NE k=NL 2 / H.rdA + 2 / N.RdC = 0 (4.16) j=l A k=l C ^ where Ag = element subdomain, R = residual which results from the use of C in Eq. 4.15 and 5 = variable length along the k^ boundary. The governing equation used by Ariathurai (1974) was Eq. 3.5 with the off-diagonal dispersion coefficients, and D„„, equal to zero. This formulation was modified to include the terms involving D^^y and Dyj^. The following development is for the modified formulation incorporated in CSTM-H. Substituting the expressions for r and R into Eq. 4.16 yields j=NE ac ac ac a ac 2 //. N . { — + u — + v (D — + j=i '^a at ax ay ax ^^ax 197 ac a ac ac D — ) (D — + D — ) - S } dxdy + ^ydy dy >^ydx yyay k=NL 2 (q^ + q- + q^) dC = 0 k=l (4.17) The term Q = _L + S in this equation is taken to be an instantaneous at constant. This approach, which transforms Eq. 3.5 into an elliptic equation, results in a more efficient computational scheme. Using Pick's law, the fluxes in Eq. 4.15 are given by dc ac ac ac (D — + D — ) n + (D — + D — ) n XX 3^ xy^y X yx^^ yy^^ y (4.18) where n^, ny = x- and y- components of the outward normal to the boundary surface of Ag. The second derivative terms in Eq. 4.17 require continuity of the first derivatives to insure convergence. This would involve solving for the unknown first derivatives of C. To avoid this added complication, the second derivatives are transformed to first derivatives using the divergence theorem. The first term in Eq. 4.16 may be expressed in vector form for one specific element as n (v'vc - v(Dvc) + Q) dv = 0 (4.19) 198 where -V = solution domain, D = sediment dispersivity tensor, and V = two-dimensional vector operator. The dispersive flux vector F is equal to DVC. Thus, the second term in the integrand of Eq. 4.19 becomes NV(DVC) = NVF = V(NF) - (VN) * F (4.20) The divergence theorem states that V • (NF) dV = NF • ndS (4.21) where n = outward normal to the surface S of the domain -V. Substituting Eqs. 4.20 and 4.21 into Eq. 4.19 gives VN L_N (V • VC + — • (DVC) + 0) dV - / N DVC • ndS (4.22) ^ N ^ Substituting Eqs. 4.18 and 4.22 into Eq. 4.17 yields for locally constant dispersion coefficients A. A j=NE ac ac ^1 ac 2 //, [N (Q + u— + V—) + (D^ — + j=i e ax ay ax ^^ax ac K K 2 N. qj dC = 0 (4.23) I<=1 ^ " 19? This equation may be expressed for a singular element by the element matrix differential equation a{c} [k] {C} + [t] + {f} + Cb] {O = 0 (4.24) at where [k] = element steady-state coefficient matrix [t] = temporal matrix {C} = vector of unknown nodal concentrations {f} = element source/sink vector [b] = boundary or element load matrix. Equation 4.24 is evaluated for each element with the element load matrix [b] = 0 for interior elements. The element coefficient, temporal and boundary matrices are given in Appendix B. The element coefficient matrix is modified to account for prescribed nodal boundary conditions by eliminating the row and column corresponding to that nodal unknown. For those boundary nodes at which no boundary conditions (i.e. concentrations or fluxes) are prescribed, the normal concentration flux across that node is set equal to zero. Next, the element matrix differential equations (Eq. 4.24) are assembled to form the system matrix differential equation a{c} [K] {C} + [T] + {p} + [B] {c} = 0 (4.25) at m where all the matrices and arrays are the system equivalents of those given in Eq. 4.24. Rearranging Eq. 4.25 and replacing the partial derivative with a finite difference gives [T] { + [K] + [B]} {c} + {F} = 0 (4.26) At Applying a Crank-Nicholson type representation to temporally discretize this equation gives {—+ Q[K]n+l + 0[B]"-'l} {C}"+1 = {—. [(l-Q)rK]" - At At (i-e)[B]"}{c}" + eiF}"'-^ + {i-e){F}" (4.27) where 0 = degree of implicitness (0 = 1, fully implicit; e = 0, fully explicit), and the superscripts n and n+1 indicate the values of the arrays and vectors at the current time step (t = nAt) and at the next time step (t = (n+l)At), respectively. The value of e is specified by the user. For stability reasons, 9 should be greater than or equal to 0.50. Using the specified initial and boundary conditions, Eq. 4.27 is solved for the NP-NBC unknown nodal concentrations at t = (n+1) At, where HP = number of nodes in the system and NBC = number of boundary nodes with specified boundary conditions. The method used to solve Eq. 4.27 is discussed next. 201 4.4.4. Equation Solvers CSTM-H contains two algorithms which solve Eq. 4.27. One algorithm uses the Gaussian elimination technique and the other uses the frontal solution program for unsymmetric matrices developed by Hood (1976). The frontal algorithm was developed specifically for applications of the finite element method to boundary value problems. Although it is based on the Gaussian elimination technique, it has advantages over the conventional banded matrix techniques in that computer storage requirements and computation times may be considerably reduced in certain applications. This is especially true for large systems which might have 1000 or more variables. For small systems (e.g. with 200 or less nodes), there is no appreciable difference in the computation time required by the two algorithms. The user of CSTM-H specifies whether the band or frontal method is to be used in solving Eq. 4.27. It is necessary to create a temporary data file on a disk when the frontal algorithm is used. 4.5. Convergence and Stability The accuracy of the numerical scheme used in CSTM-H has been investigated in detail by Ariathurai (1974) and Ariathurai et al. (1977). These authors reported that rapid convergence to the exact analytical solutions was achieved for the numerical formulation for the one-dimensional, transient heat conduction problem with and without radiation, the one-dimensional, steady-state and transient convection- diffusion problem, and for the two-dimensional Laplace equation. The results from these convergence tests also indicated that the combination of the unconditionally stable finite element formulation 202 used to solve the spatial problem and the unconditionally stable Crank- Nicholson type finite difference formulation used to solve the temporal problem is as well unconditionally stable. However, these tests revealed that instabilities might still occur when the Peclet numbers (ratio of advection to dispersion, i.e. ul/d^, where u = flow velocity, = dispersion coefficient and L = system longitudinal dimension) become either too large (greater than = 100) or too small (less than - 10' ). For too large Peclet numbers, smaller time-steps must be used to improve the accuracy of the numerical scheme (Ariathurai et al . . 1977). Too small Peclet numbers rarely ever occur for typical flow conditions in estuaries, and therefore associated roundoff errors, which can lead to instabilities, should never be a problem in modeling such systems. However, spurious results caused by roundoff errors were encountered in simulating laboratory depositional experiments with CSTM- H. This problem was eliminated by using double precision arithmetic in the model. No instability problems were encountered in modeling a prototype system with CSTM-H using single precision arithmetic. A value between 0.5 and 1.0 should be used for the degree of implicitness in order to insure a stable numerical scheme in time. High values of this parameter result in a smoother, though no more accurate solution than values near 0.5. For modeling tidal bodies of water, a time step of the order 10-30 minutes should be used. The element sizes should be chosen such that the required detail is obtained in critical areas of concern. CHAPTER V MODEL VERIFICATION AND APPLICATION 5.1. Introductory Note The purpose of this chapter is twofold: 1) to verify the CSTM-H model by demonstrating it's capability of predicting cohesive sediment transport processes, and 2) to apply the model to a two-dimensional, prototype scale body of water. The first objective is achieved by using CSTM-H to simulate five different laboratory sediment transport experiments and comparing the measured and predicted results. The second objective is achieved by using the model to simulate sedimentation in a coastal marina. 5.2. Laboratory Experiments A total of four laboratory experiments were conducted at the Coastal Engineering Laboratory during this study. Three of the four were conducted in a 18.3 m long, 0.61 m wide and 0.91 m deep recirculating flume and the fourth in a 0.2 m wide, 0.46 m deep and 0.76 m mean radius rotating annular flume. The experiments in the recirculating and annular flumes are described in Sections 5.2.1 and 5.2.2, respectively. The results of the model simulations of these experiments are presented in Section 5.2.3. 203 204 5.2.1. Recirculating Flume Experiments 5. 2.1. a. Facilities The 18.3 m long flume in which three erosion-deposition experiments were conducted is shown in Fig. 5.1. A schematic diagram of this flume is given in Fig. 5.2. The main components of the flume are the fol lowing: 1) A chamber located at the upstream end of the flume into which water from the recirculating pipe is discharged. A flow straightening device was placed in the entry chamber just downstream of the water discharge pipe. 2) The flume has a steel plated bottom and back wall with glass panels along the front side. Two electric jacks located 1.5 m downstream of the entry chamber can be used to tilt the flume to a maximum slope of 0.02. 3) An underflow tail gate at the downstream end of the flume. 4) A 1.83 m long, 0.91 m wide and 1.2 m deep transition tank, into which the water in the flume flows. 5) A centrifugal pump with a maximum capacity of 0.164 m^/s. 6) A discharge control valve. 7) A 0.2 m diameter, 15.3 m long PVC return pipe. 8) A 0.33 m wide, 4.9 m long width restricting apparatus was placed in the upstream half of the flume along the back wall (Fig. 5.1). The central constant width section was 1.52 m in length, and the two curved end sections were 1.69 m in length. This apparatus was constructed out of 0.31 cm plywood. Two coats of fiberglass resin were applied to the bottom and front and back sides to minimize swelling of the plywood. Concrete blocks were used for ballast Fig. 5.1. Downstream View of Recirculation Plume. Width Reducing Device is Shown on Right Side of Flume. 206 Q. aj c > c o c o Q, OJ w O 3 a m "w o 9- Q- > cr o q: a. 5 ■— 0, O — CM rO T m U1 a. a 3 -3 to « OJ o So g«J.S^-T3_ = -2 o ° S :t: if a u.Q.a.:SQQ </3(— CvJ CO s- (T3 E =3 CD +-> fO O s- o <u ■r- Q y 'I— u C/1 CVJ U1 9) 207 (see Fig. 5.1), and silicon and duct tape were placed along the submerged edges of the apparatus to prevent sediment from seeping behind and/or underneath the apparatus. This width restricting apparatus was placed in the flume in order to achieve a region of higher flow velocities. A 12 mm diameter, 0.9 m long PVC pipe, capped at both ends, with 1.5 mm diameter holes 5 cm apart, and connected by a rubber hose to an air compressor, was placed width-wise across the bottom of the transition tank. The turbulence generated by the jets of compressed air from this pipe served to increase the turbulence in the tank. This helped to minimize the possibility of any sediment depositing in the tank, which had a cross-sectional area approximately three times larger than that of the flume, and therefore a much lower shear stress. 5.2.1.b. Instrumentation Velocity measurements were made during the experiments using a Kent miniflow current meter (Model Number 265). The current meter consists of a 1 cm diameter, 5 bladed impeller attached to a circular frame, which is itself connected to a 48 cm long shaft (Fig. 5.3). There is a 0.1 mm clearance between the tip of each blade and the base of the shaft. Inside the shaft is an insulated gold wire. An electrical impedance between the gold wire and the shaft is changed by the rotation of the impeller in a conductive fluid (Wang, 1983). This change in the impedance modulates a 15kHz carrier signal, the strength of which is indicated on the needle dial on the monitor box. The variance in the impedance is a function of the impeller's rotation rate, which is in 208 Fig. 5.3. Kent Mini-Flow Current Meter. Calibration from Supplier n I I ! \ ! \ ! ! ! ^ ! ^0 2.0 4.0 6.0 8.0 10.0 INDICATOR READING (Hz) Fig. 5.4. Calibration of Kent Mini-Flow Current Meter. 209 turn a function of the fluid's flow velocity. The current meter was calibrated by Dixit (1982) in the Department of Civil Engineering Hydraulic Laboratory's flume and by the author in the rotating annular flume at the Coastal Engineering Laboratory. Figure 5.4 shows the good agreement obtained between the calibration provided by the manufacturer and those obtained by Dixit and the author. The current meter shaft was clamped to a vertically positioned point gage, which in turn was bolted to a horizontally positioned point gage (Fig. 5.5). As seen in this figure, the horizontal point gage was bolted to one side of a wooden cart. This set-up permitted exact (within the 1 mm accuracy of the two verniers) horizontal and vertical positioning of the current meter inside the flume. The wooden cart was placed on top of the two 2.5 cm diameter stainless steel rods which spanned the length of the flume. The cart was moved on top of the rods to the desired location along the longitudinal axis of the flume. The time-averaged elevation of the water surface above the flume bed was measured during the course of each experiment using the water surface elevation measuring device shown in Figures 5.6 and 5.7. This device, which operates on the siphon principle, consists of an electric point gage, a 5.0 cm diameter plastic tube, two valves, and two 0.5 cm diameter clear rubber hoses. The point gage was attached to the top of the tube, as seen in Fig. 5.6, and positioned such that the tip of the gage touched the water surface at approximately the tube's center. The effect of surface tension between the inner tube wall and the water inside the tube did not influence the water surface elevation at the tube's center because of the relatively large tube diameter (Wang, 1983). A 3.7 mm diameter glass tube was inserted in the unattached end 210 Fig. 5,5. Instrumentation Cart and Setup of Kent Mini-Flow Meter and Two Point Gages. 211 5.6. Electric Point Gage and Tube of Water Surface Elevation Measuring Device. 5.7. Setup of Water Surface Elevation Measuring Device (after Wang, 1983). 212 of the rubber hose. The glass tube was then inserted through a board located a few inches above the water surface in the flume. This was done in order to maintain the vertical positioning of the glass tube. A hand operated suction pump was used to start a siphon between the water in the tube and in the flume. The slow response time between the water levels in the flume and tube, caused by the small diameter and the long length of the connecting hose, resulted in a filtering of the high frequency, turbulent fluctuations of the water surface. As a result, only the time-averaged water surface elevation was measured. Two of these devices were used to measure the water surface elevation at three locations along the flume. One device continuously measured the elevation at a downstream station, while the second device monitored the elevation at an upstream station and a station in the reduced width section. As the water surface elevation can be measured at only one location at a time, the upstream elevation was measured by first closing the valve of the middle station hose and then opening the valve of the upstream station hose, and vice versa. The density profile of the sediment bed in the flume was measured using a specially designed apparatus (Parchure, 1980). A sketch of this apparatus 1s given in Fig. 5.8. To obtain a core of the bed the 2 cm diameter plastic tube is inserted in a vertical position through the sediment bed until the flume bottom is reached. Then the 15 cm diameter plexiglass cylinder with a sealed bottom is lowered concentrically around the plastic tube until it also is positioned on the flume bottom. The annular space between the 2.5 cm diameter metal tube and the outer wall of the plexiglass cylinder (see Fig. 5.8a) is filled with denatured alcohol, to which pieces of dry ice (solid carbon 213 (a) SKETCH OF APPARATUS I ZZ5 cm T 1 \ Top Cylinder IScmdia. ^ i 1 i ! Rastic Tubes d various heights, 0.95 cm dia. glued to Ite bottom plate 7.5 cm I Bottom n Cylinder P 15cm dia-*| Bottom Rate-^ (b) SKETCH OF /APPARATUS H T 15 cm _L -2 cm dia plastic tube — 15 cm dia. plexiglass cylinder — 2.5 cm dia metal tube ■Annular space for mixture of alcohol and dry ice Metal Plate" Sediment Porcelein Dish Riled with ics ojbes ^Piston with Screw Red 5.8. (a) Apparatus I for Obtaining Sediment Core; (b) Apparatus II for Sectioning a Frozen Sediment Core (after Parchure, 1980). 214 dioxide) are added. The combination of the alcohol and dry ice causes the sediment core inside the plastic tube to freeze in approximately 20 minutes. The plastic tube is then removed and placed horizontally in a second apparatus, as shown in Fig. 5.8b. The rectangular box is filled with ice cubes in order to keep the sediment core frozen. The piston with threaded rod is used to push approximately 5 mm lengths of the frozen core out of the plastic tube at a time. The ejected section of the core is brought in contact with a metal plate in front of the core (see Fig. 5.8b), which causes this 5 mm section of the core to quickly melt and drop into the porcelein dish. This process is repeated until the entire sediment core has been collected in separate dishes. The sediment 1n each dish is oven dried and weighed in order to determine the dry sediment mass in each 5 mm section of the frozen core. The method used to determine the density profile is described in the next section. Samples of the sediment-water mixture were collected at periodic intervals throughout the experiments using the water sampling device shown in Fig. 5.9. The horizontal sampling tubes have a 3.2 mm diameter and are spaced 4.0 cm apart. This device works on the siphon principle, so that when a sample is desired, the end of the rubber hose is lowered below the level of the horizontal sampling tube in the flume and approximately a 60 ml sample of the suspension is collected in a sample bottle. The volume of water contained in the hose is drained into another container, and then the 60 ml sample is collected in the bottle. This procedure was used so that a sample of the suspension which existed in the flume at the sampling time was collected, and not that at the previous sampling time. In all the experiments, two samples were collected simultaneously from the two lowest sampling tubes. 215 Fig. 5.9. Water Sampling Device. ^.1 0.01 0.001 GRAIN SIZE (mm ) Fig. 5.10. Grain Size Distribution of Kaolinite Used for the Experiments. 216 The suspension concentration of each water sample was determined using a Millipore filtering apparatus, an oven, and a Mettler balance {Model No. H80) with a ±0.05 mg accuracy. The following procedure was used: 1) Withdraw a certain volume of the suspension from the sample bottle using a 10 ml pipette. 2) Filter this sample through a pre-weighed Millipore filter paper with a 0.45 m pore diameter. 3) Place the filter paper containing sediment in an oven at 50 °C for at least two hours. 4) Weigh the dried filter paper. 5) Calculate the suspension concentration C = (mass of dried filter paper containing sediment - mass of filter paper alone) / (volume of suspension filtered). Commercial grade kaolinite was the sediment used in the three experiments performed in the recirculating flume. The particle size distribution of the kaolinite, determined by a standard hydrometer test, is shown in Fig. 5.10. The median particle diameter, as seen in Fig. 5.10, was 1 \im. The CEC value for the kaolinite given by the suppliers, the Feldspar Corporation, Edgar, Florida, was 5.2-6.5 meq / lOOgm (Dixit. 1982). Tap water was used as the fluid in these experiments. The chemical composition of this water is given in Table 5.1. A two week period was used to equilibrate the tap water and about 75 kg of kaolinite before the first experiment was conducted. 217 Table 5.1 Chemical Composition of the Tap Water (after Dixit, 1982) CI 26 ppm N03 0.07 ppm Fe 0.5 ppm K 1.4 ppm Ca 25 ppm Mg 16 ppm Na 10 ppm Total Salts 278 ppm pH 8.5 5.2. I.e. Procedure A zero bed slope was used in all three experiments. The water surface elevation measuring devices at the upstream and downstream stations were used to adjust the slope to zero. The following procedure was used to form a flow-deposited sediment bed in each of the experiments. The flume pump was started and the 75 (dry mass) of sediment was mixed at a shear stress of approximately 0.5 N/m for four hours. The flume calibration performed by Dixit (1982) was used to estimate the shear stress. The mixing was artificially enhanced by pushing a rubber wedge along the bottom of both the flume and transition tank in order to initially suspend all the sediment. After four hours of mixing, the flow in the flume was reduced to a shear stress of approximately 0.025 N/m^, which was maintained for eight 218 hours. Most of the sediment deposited during this deposition period. The non-uniform flow in the flume, which was caused by the width- reducing device and the flow under the downstream tail gate, resulted in a bed of variable thickness along the length of the flume. After the eight hour deposition period, the flow was stopped completely. The flow deoosited bed was allowed to undergo self-weight consolidation for the following consolidation times: 1^^ = 3, 84 and 240 hours for Test No. 1, Test No. 2 and Test No. 3, respectively. At the end of the consolidation period, the flume pump was started and the flow rate was slowly adjusted (i.e. increased) so that after 15 minutes the shear stress in the flume was approximately 0.02 N/m^. This same flow rate was maintained for an additional 45 minutes. This procedure was followed in order to resuspend the sediment which deposited during the consolidation period in the recirculating pipe, which has a flow cross-section approximately four times smaller than the flume and therefore a shear stress approximately 16 times greater than that in the flume, without resuspending any of the sediment in the flume. The suspension concentration which existed in the flume after this one hour period was regarded as the initial concentration, Cq, at the start of each experiment. was determined by water samples collected at the end of the one hour resuspension period. In addition, the bed surface elevation which existed at the end of the one hour period was measured through the front glass panels every 0.91 m along the flume test section. These measurements were used as initial bed conditions in the model simulations. The approximate shear stress history for the three experiments are shown in Fig. 5.11. This figure may be interpreted as follows. In Test 219 C.I5f 0 0.05 on 1 \ ^b2 r. b! ■^0 J L I I I 2.0 4.0 6.0 TIME (hours) (a) TEST NO. I 80 0.2- cn 00 LU QC t- cn cr < LU X cn Q LU CO 0.1 bi 0.0, 0 J L O.i O.QL 0 n I — 7=— r 1 r 2.0 4.0 60 TIME (rxDurs) (b) TEST NO. 2 80 0.2i — -^FT^ \ \ \ r ^b. J L 2.0 4.0 6.0 TIME (hours) (c) TEST NO. 3 80 ICO Fig. 5.11. Shear Stress History for Experiments in the Recirculating Flume. 220 ^ ^'''dc ^ hours), the approximate shear stress in the unrestricted width sections of the flume was equal to 0.06 N/m^ for the first two hours, then it was increased to 0.12 N/m^ for the third and fourth hours, and finally it was decreased to 0.033 N/m^ for the final five hours. In Test No. 2 (T^^ = 84 hours), the approximate shear stress was 0.075 N/m^ for the first 3.5 hours, 0.17 N/m^ for the next 2.5 hours, and 0.026 N/m^ for the final five hours. In Test No. 3 (T^^ = 240 hours), the approximate shear stress was 0.17 N/m^ for the first two hours, 0.026 N/m^ for the next five hours, and 0.075 N/m^ for the final two hours. It took approximately two minutes to change the flow rate in the flume to the new shear stress value and establish steady flow conditions. During each of the constant shear stress time intervals, water samples were collected at sampling Station A (see Fig. 5.12) from the two lowest sampling tubes at 0, 2, 5, 10, 20, 30, 40, 50, 60, 75, 90, 105, 120 minutes, etc. after the shear stress was changed. In order to determine if any spatial (i.e. longitudinal and transverse) variability in the suspension concentration occurred during the experiments, water samples were also taken at alternate sampling times at Station B during Test No. 2 and at Station C during Test No. 3. Figure 5.12 shows the reach of the flume used in each experiment, and the locations of the three water sampling stations (A, B, C), the three water surface elevation measuring stations (D, E, F) and the three velocity measuring stations (G, H, J). The water surface elevation was monitored periodically at each of the three measuring stations during each constant shear stress time interval. The bed thickness was measured every 0.91 m along the length 221 £ O -a c as as V) o +J >1 ul -a O) CO Q O (/I c/) != O o a. • (T3 fO CO T3 01 CD £_ E 3 •r- l/l Qj O) (/I I s: oj to C •l- E O M- O T- O •r- +J S- +-> fO Q. n3 > +J Ol >, 00 I— +-) LU -r- cn CJ E 0) O •1- O r— r— fO QJ Q-H- > E ^ (0 3 1 — u S_ S_ -r- 4-> +-> S- ra rd O) -a CO 3 IZ CD o c o •r- -t-3 nS 4J 00 4-> c i cu s- =s Wl <0 0) CM I— I LO cn 222 of the flume's test reach at least once during each constant shear stress time interval. In addition, the water temperature was measured every hour over the duration of each experiment. The vertical velocity profiles at five lateral positions at Stations G and J, and at three lateral positions at Station H (see Fig. 5.12) were measured once during each constant shear stress interval using the Kent mini flow meter. After the three experiments were completed, the identical bed formation process was again repeated. A frozen core of the sediment was obtained for the purpose of determining the density profile at each of the consolidation times (i.e. 3, 84 and 240 hours) used in the experiments. These cores were collected downstream of the width restricting device, where the bed had approximately uniform thickness. The following procedure was used to determine the bed density profile from each segmented core. The density of each 5 mm thick layer was determined by dividing the dry sediment mass in that layer by the volume. Freezing of the core caused the sediment sample inside the core to swell. A uniform correction for the swelling was made by dividing the total thickness of the frozen core by the total thickness of the original (i.e. before freezing) core, and then dividing the thickness of each frozen layer (=5 mm) by this factor. This gives the mean density in each layer, from which the density profile may be constructed. 5.2. l.d. Results Figures 5.13-5.15 show the variation of the suspension concentration with time during Tests No. 1, 2 and 3, respectively. The concentrations determined for both the lower and upper sampling tubes 223 0.0 2.0 4.0 6.0 8.0 TIME (hrs) Fig. 5.13. Measured and Predicted Suspended Sediment Concentrations for Test No. 1. 224 00' I 1 I I I I I I I I 0.0 2.0 40 6.0 8.0 10.0 TIME (hrs) Fig. 5.14. Measured and Predicted Suspended Sediment Concentrations for Test No. 2. i 225 6.0 I 4.0 H UJ u z o u o CO UJ a. CO 3 07 2.0 Station • A- Upper ^ A - Lower o C- Upper ^ C - Lower n 1 r Test No. 3 V 240 hrs Concentrations — Predicted X o Measured '•A o ^ 00 0.0 2.0 4.0 TIME (hrs) 6.0 8.0 Fig. 5.15. Measured and Predicted Suspended Sediment Concentrations for Test No. 3. 226 are shown in these figures. In addition. Fig. 5.14 shows the suspension concentrations determined for the downstream collection station (Station B) during Test No. 2, and Fig. 5.15 shows the measured concentrations at Station C during Test No. 3. As evident in these two figures, the suspension concentration varied slightly in the flume in both the longitudinal and lateral directions. The lateral gradient in C was caused by the non-uniform lateral velocity profile which existed in the flume. The longitudinal gradient was caused by the combined effects of longitudinal dispersion, erosion and/or deposition which occurred along the length of the flume. The concentration values determined from the lower sampling tubes were plotted in Figs. 5.13-5.15 only when they differed from the values at the upper sampling tubes by more than ±0.03 g/1. No significant difference (i.e. vertical gradient) in the suspension concentration was observed between the upper and lower sampling locations during the three experiments, except for the time period immediately following changes in the flow rate. This phenomenon is best exemplified by the large vertical variation in C (up to 2 g/1) during the first half hour period of deposition (from 6.0 to 6.5 hours) in Test No. 2 (see Fig. 5.14). Differential settling is the probable cause for this observed vertical concentration gradient during the initial stages of deposition. The high vertical variation in C (up to approximately 1 g/1) found, for example, during the first half hour period of erosion (from 2.0 to 2.5 hours) in Test No. 1 (see Fig. 5.13) is probably due to higher resuspension rates than vertical diffusion rates during this initial period. As the resuspension rate decreases due to increasingly smaller values of the excess bed shear (caused by 227 the Increase of the bed shear strength with depth below the Initial bed surface), the continuing vertical diffusion of sediment reduces the magnitude of this vertical concentration gradient, as seen in Fig. 5.13 from 2.5 to 4.0 hours. The depth-averaged velocity at each vertical was determined by plotting the vertical velocity profile and then integrating it using a planimeter. The water depths at Stations D, E and F were determined by subtracting the measured bed thickness from the water surface elevation above the bottom of the flume. The bed density profiles for the three experiments are given in Fig. 5.16. The applicability of the power law relationship between the dry sediment density, p, and the depth below the bed surface, z^, as given by Eq. 3.76, is apparent for the three measured density profiles. 5.2.2. Rotating Annular Flume Experiment The purpose for conducting an experiment in the rotating flume at the Coastal Engineering Laboratory was to verify that the cohesive sediment transport model developed during this study can be used to predict erosion and deposition rates in an unsteady flow field. 5. 2. 2. a. Facilities The annular flume in which the experiment was performed is shown in Fig. 5.17. This flume has the following dimensions: 0.21 m wide, 0.45 m deep and 0.76 m mean radius. The flume consists of three main components: 1) a rotating circular fiberglass channel which holds the sediment-water mixture, 2) an annular ring with a slightly smaller width and the same mean radius as the channel, and 3) a steel frame and 228 p / p . 5.15. Measured Bed Density Profiles for Experiments in Recirculating Flume. 229 Fig. 5.17. Rotating Annular Flume. 230 electric motors. The ring, positioned in contact with the water surface, and the channel are rotated simultaneously in opposite directions in order to achieve a nearly uniform turbulent shear field in the channel, and to minimize the effects of rotation-induced radial secondary currents. This design and operational procedure eliminates the need for aggregate-disrupting elements such as circulatory pumps, in which very high shearing rates usually occur. The required bed shear stress is attained by adjustment of the rotational speeds of the channel and the ring. Four taps, located on the outer channel wall (see Fig. 5.17), are used to collect suspended sediment samples from the channel. 5.2.2.b. Instrumentation A Hewlett Packard HP-85 micro-computer with two digital-to-analog converter units were used to control the rotation rates and accelerations of the channel and the ring. The micro-computer was programmed to generate the desired flow field in the channel. The bottom sediment from Lake Francis, Nebraska and tap water with a 10 ppt solution of commercial grade sodium chloride were used in this experiment. X-ray diffraction analysis performed by the Soil Characterization Laboratory at the University of Florida revealed that this sediment is predominantly composed of montmorillonite, illite, kaolinite and quartz. 5.2.2.C. Procedure The sediment and 10 ppt saline water were placed in the channel. A 30.5 cm depth of flow was used. The sediment-water mixture was mixed at a shear stress of approximately 1.7 N/m^ for 24 hours. The flume was 231 then stopped and the suspended sediment was allowed to deposit and undergo self-weight consolidation for 40 hours. The HP-85 micro-computer was programmed to generate a uni- directional, semi-diurnal, constant depth tidal flow. The cross- sectionally averaged sinusoidal velocity-time record used in this tidal cycle experiment is given in Fig. 5.18. Because of a mechanical probl with the bearings in the channel's drive shaft, it was not possible to have a true slack period in this experiment. Though not apparent in Fig. 5.18, the flow velocity was discretized into five minute increments, during which the velocity varied as follows: during the first 30 seconds of each five minute time increment, the velocity was linearly increased or decreased to the next value (as determined by a sinusoidal velocity-time relationship), while for the remaining 4.5 minutes, a constant velocity was maintained. The program was started and run for two tidal cycles (25 hours) in order to establish quasi -steady state conditions. Water samples were collected from the mid-depth tap at the end of this 25 hours and every five minutes thereafter for 16 hours. The samples were collected just before the end of each five minute increment (i.e. before the velocity was changed). The temperature of the water in the channel was measured every hour. In addition, water was added every hour to replace that withdrawn in the water samples in order to keep the ring in contact with the water surface. em 232 233 5.2.2. d. Results Figure 5.18 shows the variation of the suspension concentration over the duration of the 16 hour experiment. Also plotted in this figure is the variation of the cross-sectionally averaged velocity. A short lag between the flow velocity and the suspension concentration is seen to occur immediately following the occurrences of minimum and maximum velocities. 5.2.3. Model Simulations The test reach of the recirculating flume was divided into the 34 element, 141 node finite element grid shown in Fig. 5.19. Zero sediment flux boundary conditions were used for the upstream and downstream flow boundaries. The depth-averaged velocities measured at Stations G, H and J were used for the nodal velocities in elements 1-2, 7-10, and 15-34, respectively. The nodal velocity vectors in the converging section (elements 3-6) of the flume and in the diverging section (elements 11- 14) were determined from continuity considerations. Likewise, the water surface elevations measured at Stations D, E and F were used for the nodal values in elements 1-2, 7-10, and 15-34, respectively. The nodal water surface elevations in elements 3-6 and 11-14 were determined by linear interpolation of the values measured at Stations D and E, and Stations E and F, respectively. The initial bed thicknesses measured every 0.91 m along the flume were used as the initial bed thicknesses for the even-numbered elements. The initial bed thickness of the ith_ odd-numbered element was assumed to be equal to that of the (i+l)th even-numbered element, for i=l, 33. The dry mass of the sediment forming the bed in each element at the start of each experiment was 234 (r 5^ 6 7 8 9 10 II / 12 /,4 15 16 17 IB 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Fig. 5.19. Finite Element Grid of Recirculating Flume; Distorted Sketch - Width: Length = 4.1: 1.0. 235 determined using the measured bed thickness and measured bed density profile. The dry mass in each element and the measured bed density profile were read into CSTM-H in order to form the initial bed for each experiment. The parameters which characterize the erosional and depositional characteristics of kaolinite in tap water, determined by Dixit (1982) and Mehta (1973), respectively, and the consolidation parameters given in Chapter III, Section 3.7.1 were used in CSTM-H to simulate the three experiments in the recirculating flume. The dispersion coefficients were calculated using the dispersion algorithm. A two minute time step was used in the model simulations. Comparisons of the predicted and measured suspension concentrations for the three experiments are shown in Figs. 5.13-5.15. Satisfactory to good agreement is observed in all three experiments. A discussion of these results is given in Section 5.4. A four element, 23 node straight grid was used to represent the annular flume in simulating the tidal cycle experiment. The length of this grid system was set equal to the circumference of a circle with a radius equal to the mean radius of the flume. The suspension concentration at the "downstream" flow boundary was used for the "upstream" boundary condition in order to represent they were the same boundary. The velocity record shown in Fig. 5.18 and the erosional and depositional characteristics of Lake Francis sediment reported by Mehta et al_. (1982a) and Hayter and Mehta (1982) were used in CSTM-H. A 2.5 minute time step was used in the model simulation. A comparison of the predicted and measured suspension concentrations is as well shown in Fig. 5.18. A discussion of the simulation result is given in Section 5.4.. 236 5.3. Simulation of WES Deposition Experiment A deposition experiment conducted in the 99.7 m long, 0.46 m deep and 0.23 m wide non-recirculating flume at the U.S. Army Corps of Engineers Waterways Experiment Station, Vicksburg, Mississippi was simulated with CSTM-H. A schematic diagram of this flume is shown in Fig. 5.20. A detailed description of the flume is given by Dixit etal. (1982). The purpose of this experiment and three other experiments conducted in the 99.7 m long flume was to investigate the phenomenon of sediment sorting in the longitudinal direction, the occurrence of which is well documented in muddy estuaries (Edzwald et al_. , 1974; Dixit et al_'. 1982). The sediment used in this experiment was the commercial grade kaolinite described in Chapter III. The fluid used was tap water with a chloride concentration of 18 ppm. pH = 7.8 and sodium adsorption ratio SAR = 2.07 (Dixit etal., 1982). The procedure used in this deposition experiment was the following. A 100 g/1 sediment-water slurry was prepared by mixing the kaolinite and water for three hours. The slurry was injected at a specified rate into tap water which flowed from the headbay into the flume (see Fig. 5.20). The slurry injection rate was regulated so that the resulting suspension had a concentration of approximately 4.0 g/1. The flow rate was held constant for three hours, after which only the clear water flow was maintained until the sediment cloud had passed out of the flume. The thickness of the deposited bed was measured 15 minutes after the end of the experiment. Vertical and horizontal velocity profiles and the water surface elevations were measured at several stations along the length of the flume. In addition, vertical 237 = 1 a> ^ o a i_ p (U 3 o o 0) c B X LxJ i- Q. 3 +J 0) 00 o CM C CO O <Ti •1— r-H ■M fC +J . C "—I <D fdl s- (d| Q- q; -1- X o •■- •I- Q +-> ra S- E CD OJ +J ^ <+- CO > o 238 suspension concentration profiles were measured at several stations along the centerline of the flume every 15 minutes during the course of the experiment. The mean flow depth, mean velocity and the bed shear stress during the experiment were 0.162 m, 0.091 m/s, and 0.033 N/m^, respectively (Dixit etal_., 1982). This experiment was simulated using a 10 element, 53 node grid and a four minute time step. The stated flow conditions, the depositional properties of kaolinite in tap water found by Mehta (1973) and the consolidation properties of kaolinite given in Chapter III, Section 3.7.2 were used in CSTM-H to perform the simulation. For the upstream boundary condition, the suspension concentration was set equal to 4.0 g/1 for three hours, and then set to 0.0 g/1 for the remainder of the experiment. A zero concentration flux was used for the downstream boundary condition. The model simulation was continued until the maximum predicted concentration in the flume was 0.04 g/1 (1% of the initial injection concentration). Figure 5.21 shows the comparison between the measured and predicted bed thickness profile at the end of the experiment. A discussion of the results of this simulation is given in the next section. 5.4. Discussion of Results The model simulations of the three experiments in the recirculating flume yielded good agreement between the measured and predicted temporal variations in the suspended sediment concentration, as seen in Figs. 5.13-5.15. In Test No. 1, it is apparent from the measured concentration-time record that during the two erosion intervals (i.e. 239 i r I I I cu ^ Jo 0. O O 00 O o CD O IT) O ro 8 _o OJ 00 (X) o — d o d (ujo) SS3N>iDIHl 039 E LlJ O 9 < LU O z 240 during the first four hours) the bed was primarily redispersed (mass eroded) by the flow-induced bed shear stress. Deviations between the measured and the predicted suspension concentrations (i.e. between the observed and the predicted rates of erosion) occurred because the consolidation algorithm in CSTM-H stipulated that the bed, for which T^^ > 2 hours, would undergo the slower process of resuspension when subjected to an excess shear. Even though the predicted erosion rate during the first hour of each erosion interval was less than that reflected in Fig. 5.13, good agreement was achieved between the total mass of sediment that was measured and predicted to erode over the duration of each interval. Nevertheless, the fact that the bed was apparently redispersed when subjected to the given bed shear stress points out a limitation of the erosion-consolidation algorithm in CSTM-H and an area for future research. The deposition stage in Test No. 1 was in Range IB (see Fig. 3.48) where the settling velocity decreases with a decrease in the suspension concentration according to W3 a c\ with n = 1.33. The good agreement obtained between the measured and the predicted concentrations in this interval (see Fig. 5.13) was obtained by adjusting the value of the proportionality constant (kJ in Eq. 3.51b) between and c". In Test No. 2 with T^^ = 84 hours, the measured suspended sediment concentrations again indicate that the top layer of the bed was apparently redispersed during the first erosion step (see Fig. 5.14). A slightly better agreement between the measured and the predicted concentration was achieved during this first step by increasing the value of the parameter a (see Eq. 3.13) in the top bed layer from the value obtained by Dixit (1982). 241 The measured concentrations during the second erosion interval in Test No. 2 indicate that the bed eroded aggregate by aggregate (resuspended) after the initial five minute interval (from 3.50 hours to 3.58 hours), during which redispersion occurred. Because the bed was resuspended during all but the first five minutes of this interval, a better agreement was obtained between the measured and the predicted concentrations, and therefore also between the actual and the predicted rates of erosion. However, as apparent in Fig. 5.14, the predicted concentrations were consistently lower than the measured concentrations during the second erosion interval. A better agreement could have been obtained by increasing the values of the aggregate erosion rate (see Eq. 3.13). Good agreement was again achieved during the deposition stage in Test No. 2, although the predicted deposition rate was slightly higher than that observed during the last two hours of the experiment. This resulted in a lower predicted than observed concentration at the end of the experiment. In Test No. 3 with T^^ = 240 hours, resuspension occurred during the first erosion interval after the first 10 minutes during which redispersion apparently occurred. The redispersion resulted in higher observed than predicted suspension concentrations for the first hour. Fairly good agreement was achieved during the second hour. Good agreement was obtained between the measured and the predicted concentrations during the five hour period of deposition in Test No. 3 (see Fig. 5.15), although from 2.5 hours to 5.0 hours the predicted deposition rates were slightly less than the observed rates, while during the last hour of the deposition interval (from 6.0 hours to 7.0 242 hours) the predicted deposition rates were slightly greater than the observed rates. The measured suspension concentrations during the second erosion step in Test No. 3 (from 7.0 hours to 9.0 hours) indicate that the top portion of the sediment which deposited during the previous five hours was eroded in mass by the increased bed shear stress. Redispersion was simulated by CSTM-H to occur during the first time step of increased bed shear. The occurrence of redispersion is not evident in Fig. 5-15 because of the extremely thin unconsolidated new deposit (UND) layer (and therefore small quantity of dry sediment mass which forms this layer) used in this simulation. The subsequent resuspension predicted by CSTM-H yields a slightly greater eroded sediment mass than measured. Figure 5.18 shows the comparison between the model simulation of the tidal cycle experiment in the rotating flume and the measured suspension concentrations. A good agreement between the predicted and measured concentrations is seen. The most noteworthy differences are that the measured concentrations lag the predicted values by approximately 20 minutes at the times of maximum concentrations (which correspond to the times of maximum or peak tidal flow velocities), while the predicted concentrations lag the measured values by approximately 20 minutes at the times of minimum concentrations (which corresponded to the times of minimum flow velocities). Thus, assuming the velocity-time record used in the simulation is correct, the predicted periods of erosion and deposition are approximately 40 minutes longer and shorter than, respectively, the observed periods. There are at least four possible explanations for this difference in the predicted and observed periods of erosion and deposition. 1) A time lag between the change in 243 the rotation rates of the ring and the channel of the rotating annular flume and the resulting response in the flow velocity. 2) The occurrence of deposition even in an accelerating flow when the flow velocity is below a certain minimum critical value. 3) The decrease in the flow acceleration leading up to a maximum velocity might cause a portion of the suspended sediment in the upper half of the water column to settle below the mid-depth tap where the water samples were obtained. This would result in a decrease in the suspension concentration as determined from the mid-depth sampling location, and therefore seem to indicate that deposition had occurred, even though the sediment might have just settled below mid-depth and not have actually deposited on the bed. 4) The 20 minute lag between the onset of accelerating flow and the increase in the measured suspension concentration (as observed at about 6.0 hours in Fig. 5.18) might have been caused by the lag time between the erosion of sediment and the vertical diffusion of this sediment up to the mid-depth sampling location. The merits of the second and fourth possible explanations are questionable in light of the fact that no time lag is observed between the measured and predicted concentrations at 12.5 hours in Fig. 5.18. The first possible explanation is not very plausible since the response time of the water in the channel to changes in the rotation rates of the ring and the channel is generally of the order of one to three minutes. The explanation given (number three) for the observed lag between the predicted and measured concentrations at maximum flow velocities needs to be investigated in the future by repeating this experiment and taking water samples at several locations over the flow depth during the latter half of accelerating flows. 244 As observed in Fig. 5.18, the measured increase in the amount of sediment eroded per half tidal cycle (i.e. during the two accelerating flow periods per tidal cycle) was predicted fairly accurately by CSTM- H. The explanation for this slight increase in eroded sediment is the following. After the top bed layer is eroded, the shear strength of the now exposed lower bed layer decreases as the bed surface swells in response to the removal of the overburden pressure. This phenomenon is simulated in CSTM-H by changing (decreasing) the bed shear strength at the new bed surface to the value of that existed at the bed surface at the end of the previous time step. Thus, as the experiment continues, the bed shear strength at the various depths (below the initial bed surface) to which the bed is eroded becomes slightly less, which of course increases the susceptibility of the exposed sediment to erosion. This slight decrease of the bed shear strength of the surface due to swelling is greater, on a short term basis, than the increase of the shear strength due to consolidation. The fact that CSTM-H simulates the slight increase in the amount of sediment eroded per time step with good accuracy, as seen in Fig. 5.18, is an indication that this representation of the decrease in i^ at the bed surface is realistic. The simulations of the three experiments in the recirculating flume and tidal cycle experiment in the rotating flume have verified the predictive capability of CSTM-H. The ability of CSTM-H to model the longitudinal sorting process which occurred in the deposition experiment in the flume at WES is discussed next. As apparent in Fig. 5.21, satisfactory agreement was achieved between the measured and predicted deposit thicknesses in the lower 65 m reach of the WES flume. The measure thickness in the first 35 m 245 possibly shows the influence of longitudinal sorting, which results in a variation in the rates of deposition of the suspended aggregates along the flume. A causative factor of this phenomenon is thought to be differences in the composition of the particles which form the aggregates (Dixit etal_., 1982). CSTM-H is not capable of simulating longitudinal sorting because the parameters (specifically tgQ and which characterize the log-normal depositional law are assumed to be spatially invarient. However, if the relationships between t^Q and cr^ with distance along the flume were known, it would be possible to incorporate these into the deposition algorithm, and thereby have the capability of predicting the effect of longitudinal sorting on the rates of deposition. When CSTM-H is used to predict the sediment movement in an estuary, the variation of tgg and in both the x- and y- directions would have to be determined. The number of field and laboratory experiments that would need to be performed in order to determine the relationships tgg = tgg (x, y, t^) and = a, (x, y, , (or even tgQ " ■'^50 '^b^ ^2 " °2 "^b^^ ''^ thought to be impractical. Thus, only the relationships t^Q = t^Q [x^) and = (t^) were incorporated into CSTM-H. Another possible explanation for the smaller measured bed thickness in the upstream 35 m reach is the increased turbulence present in the flume immediately downstream of the point of injection. Increased turbulence would result in lower rates of deposition, and therefore to smaller bed thicknesses. It needs to be emphasized that the preceding discussion on the discrepancy in the upstream 35 m reach is based on the assumption that the bed thickness in this reach is approximately uniform and equal to the measured value at 25 m. 246 5.5. Model Applications The utility of CSTM-H is demonstrated by simulations of the sedimentation in Camachee Cove Marina and the suspended sediment transport in a 10 km hypothetical canal. Camachee Cove Marina is located on the Tolomato Ri ver-Intercoastal Waterway about 150 m north of the Vilano Bridge in St. Augustine. Florida. An aerial photograph of the basin is shown in Fig. 5.22. The single entrance channel to the basin is about 180 m long and 60 m wide, and has naturally sloped banks. The surface area of the basin is approximately 33.370 m^. The semi -rectangular shaped basin has approximate dimensions of 300 m in length and 100 m in width, being tapered towards the south end (Srivastava. 1983). The bathymetry of the marina basin determined from a survey conducted in September. 1982 is shown in Fig. 5.23. The local scour holes seen in this figure were probably caused by boat propellers. The bulkhead of the basin is made of concrete sheetpile. The sediment material in the basin was found to have eight percent of organic matter by weight, and a median particle diameter which varied from 10 m in the northern end of the basin to 40 m in the center of the basin. Such a distribution is not surprising since the largest particles would be expected to deposit in the central wide section of the marina. The hydrographic and sediment data required to model both the predominantly tide-induced circulation and sedimentation in the basin were collected by the Coastal Engineering Laboratory at the University of Florida. A detailed description of the field study is given by Srivastava (1983). The main findings of this hydrographic study were the following: 1) The tide is semi-diurnal, with a mean range of 1.4 247 Fig. 5.23. Bathymetry of the Entire Basin Obtained in September, 1982. 249 m. 2) The maxlinum velocity in the entrance is approximately 2.4 cm/s. 3) The average suspended sediment concentration was 6.2 tng/1 for the months of July and August, 1982. and 6.8 mg/1 for November, 1982 through July, 1983. The hydrodynamic modeling was performed using the two-dimensional, depth-averaged finite element flow model RMA2 (Norton et al .. 1973). The finite element grid used in modeling both the tidal flow and cohesive sediment transport is shown in Fig. 5.24. The results from the flow modeling as well as the required sediment data (e.g. suspension concentration-time record at the entrance, sediment settling characteristics) given by Srivastava (1983) were used in modeling the sedimentation in the marina using CSTM-H. The results from this modeling effort are shown in Fig. 5.25, which shows contours of the predicted amount (thickness) of sediment deposition in centimeters per year. The mean depth of the marina is predicted to reduce by 8.6 cm/year, which is 42% less than the measured 14.8 cm/year (Srivastava, 1983). The measured sedimentation rate was obtained by comparing bathymetric surveys conducted in March, 1980 and September, 1982. The observed deposition pattern is not unexpected, as the greatest amount of sediment deposition would be expected in the central wide portion of the basin where the flow velocities are considerably lower than in the relatively narrow channel. Three limitations of this modeling effort, and the probable reasons for the difference between the measured and predicted deposit thickness, are discussed next. The predicted sedimentation rate is representative of fair weather conditions only, in that the tide and suspended sediment concentration at the marina entrance (which were used for the boundary conditions in 250 251 252 the flow and sediment transport model, respectively) were measured during fair weather. It is believed that storm events may enhance the rate of sediment intrusion into the basin. This is attributable to the probable increase in concentration of suspended sediments caused by the erosion of more sediment by the storm agitated exterior body of water (i.e. the Tolomato River). The water samples collected over the depth of flow in the entrance channel during the field study, from which the depth-averaged suspension concentrations were determined, were only collected down to 30 cm above the bottom. Thus, the proportion of the suspended load, which may be considerable, that is transported into and out of the basin in the bottom 10% of the flow depth was not accounted for in the boundary conditions used in the cohesive sediment transport model. The influence of horizontal circulation due to shear flow at the entrance to the marina, as well as that due to wind-induced vertical circulation was not accounted for in the flow modeling. Thus, only the advective and dispersive transport of suspended sediment due to tidal flow in the marina was accounted for in the sediment transport modeling. The combination of these three factors, which were not accounted for in this modeling effort, are felt to be the reason why the predicted sedimentation rate was 42% less than the measured rate. The dimensions of a 10 km long hypothetical canal are shown in Fig. 5.26. The canal was divided into nine elements and 48 nodes, with the length of elements 1 - 3 equal to 833 m and that of elements 4 - 9 equal to 1250 m. The canal was assumed to have a uniform bottom roughness, as quantified by a Mannings coefficient of 0.02, which is a reasonable value for a straight natural waterway with a muddy bottom. The depth 254 and the mean velocity at nodes 1. 2 and 3 were taken to be 5.0 m and 0.5 m/s respectively. The velocities and water depths at nodes 4 - 48 were evaluated using the conservation of energy and mass equations for an open channel. The total drop in the water depth over the 10 km distance due to frictional resistance and the gradual enlargement in width at element 5 was determined to be 0.16 m. The initial suspension concentration in the canal was taken to be 0.0 g/1. The following boundary conditions were used: nodes 1, 2 and 3: C(t) = 0, and nodes 46, 47 and 48: 9c{t)/ax = 0. The upstream (i.e. nodes 1. 2 and 3) boundary condition states that no suspended sediment was transported into the canal from upstream sources, while the downstream (i.e. nodes 46, 47 and 48) boundary condition stipulates that the longitudinal flux of suspended sediment across the downstream boundary was zero. In elements 1 - 4. an initial, partially consolidated Lake Francis sediment bed 0.17 m in thickness was assumed to exist, while in elements 5 - 9, no initial bed was present. Erosion of the initial sediment bed occurred in elements 1 - 4, while deposition of the sediment suspended in the first four elements occurred in elements 5 - 9. The suspension concentration-time record for elements 4 and 5 are shown in Figs. 5.27 and 5.28 for salinities of 0, 1, 10 and 35 ppt. As evidenced by the over three-fold decrease in concentration between Figs. 5.27 and 5.28, a high percentage of the suspended sediment deposited in element 5. Also observed in these two figures is a reduction in the quality of sediment suspended with increase in the salinity. This observation follows from the previously described effect of salinity on the rate of erosion. Also apparent is the small effect of salinity on the rates of deposition. In conclusion. 255 Fig. 5.27. Predicted Suspended Sediment Concentration-time Record for Element 4 in Hypothetical Canal. 256 TIME (Hrs) 5.28. Predicted Suspended Sediment Concentration-time Record for Element 5 in Hypothetical Canal. 257 these simulations demonstrate the significant influence of salinity on the transport rate of cohesive sediments for salinities less than about 10 ppt, and the diminished influence of salinity for salinities greater than 10 ppt. 5.6. Model Limitations A two-dimensional, depth-averaged model such as CSTM-H can strictly be applied only to estuaries, harbors and basins (such as marinas) where the horizontal dimensions of the water body are at least one order of magnitude greater than the vertical dimension. Applications to partially mixed water bodies or especially to highly stratified water bodies should be made when only rough estimates of some sedimentary process (e.g. shoaling rate) are required. Currently the model has the capability of simulating the movement of only one constituent (e.g. cohesive sediment, water temperature, or algae, provided the source/sink expressions for a nonconservati ve constituent are known). It is possible, however, to modify the model so that any number of constituents may be incorporated. Probably the main "limitation" of a model arises from three sources: 1) insufficient data, 2) poor quality of data and 3) limitations of the hydrodynamic modeling. The first two sources are attributable to the fact that, owing mainly to time and cost consideratins, all the bathymetric, hydraulic and sedimentary data required for use in such a model are rarely, if ever, measured and/or collected in the body of water being modeled. In addition, the quality of the data is often questionable. Data requirements and the field collection and laboratory testing programs required to obtain these data are briefly described in Appendix D. 258 The third source is itself often the result of the first two, inasmuch as progress has been achieved in the past two decades in modeling estuarial hydrodynamics (Leendertse et al_. , 1973; King et al., 1973; Liu and Leendertse, 1978). The importance of experience in effectively using the model cannot be over emphasized. Experience gained through knowledge of the physical systems being modeled and repeated applications of the model will enhance the user's ability to choose the proper values of the various parameters, e.g. time-step size. The user will also gain the ability to anticipate the effect of changing the value of a particular parameter by a certain percentage on the model solution (i.e. model sensitivity). 5.7. Model Applicability 5.7.1. Water Quality Problems The model can be used to assist in the performance of the following water quality related computational tasks: 1. ) Assessment of the disposition of dissolved and sorbed pollutants, possibly either transported to an estuary or harbor by stormwater runoff or released into these water bodies by nearby industries, and their effect on the receiving waters and the aquatic ecosystem therein, when linked with a particulate contaminant transport model that contains a sorption submodel (Onishi and Wise, 1979). 2. ) Prediction of the effect of reduced sediment inflows to estuaries, caused by upstream water storage and subsequent use, to ascertain the degree of waste water management required to control estuarial water pollution. 259 3.) Prediction of the limitation of sunlight penetration in estuarial waters resulting from high turbidity levels which, in turn, are caused by high concentrations of suspended sediment. This reduced light penetration can cause the algae multiplication rate to decrease significantly, and thus affect the entire aquatic ecosystem. 5.7.2. Sedimentation Management Problems The model can be used as a tool to help solve the following sedimentation problems: 1. ) Prediction of the movement of dredged material released in open waters in order to estimate the effect of the disposal at a given location in the water body on the shoaling rates elsewhere, and in particular in the dredged area. 2. ) Selection of harbor sites in estuaries and bays where shoaling is minimized. 3. ) Prediction of changes in the sedimentary regime that may occur as a result of a proposed change or development of an estuary or harbor, such as the dredging of new navigation channels and the possible change in the salinity field (e.g. further inland intrusion) caused by the proposed change. 4. ) Estimation of shoaling rates and maintenance dredging requirements in areas of very low flow such as marinas, harbors and docks, and recommendation of means by which shoaling rates might be minimized. 5. ) Prediction of the spatial (primarily longitudinal) variance in the shoaling and/or erosion rates, caused by varying flow conditions and salinities, along the entire reach of an estuary. CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS 6.1. Summary and Conclusions Cohesive sediments are comprised largely of terrigenous clay-sized particles. The remainder may include fine silts, organic matter, waste materials and small quantities of very fine sand. The electrochemical surface repulsive forces which act on each elementary clay particle are approximately six orders of magnitude larger than the gravitational force. As a result, the physicochemical properties of cohesive sediments are controlled mostly by these surface forces. In water with a very low salinity (less than about 1 ppt) the elementary particles are usually found in a dispersed state. A slight increase in the salinity (up to 2-3 ppt) is sufficient to repress the repulsive surface forces between the elementary particles, with the result that the particles coagulate to form aggregates. Each aggregate may contain thousands of elementary particles. Coagulation depends upon interparticle collision and cohesion after collision. The three principle mechanisms of interparticle collision in suspension are Brownian motion, internal shearing and differential sedimentation. Cohesion of elementary particles is caused by the presence of net attractive surface forces. The latter condition is caused by the increased concentration of dissolved ions, which serves to depress the double layer around each particle and allow the attractive forces to predominate. 260 261 Most estuaries contain abundant quantities of cohesive sediments which usually occur in the coagulated form in various degrees of aggregation. The transport of cohesive sediments in estuaries is strongly influenced by the coagulation behavior of dispersed sediment particles, which is controlled by the chemical composition of the suspending fluid, the hydrodynamic conditions, the concentration of suspended sediments and the physicochemical properties of the sediment. Sediment related problems in estuaries include shoaling in navigable waterways and water pollution. The mixing zone between upland fresh water and sea water, as well as areas such as dredged cuts, navigation channels, harbors and marinas are favorable sites for sediment deposition. Since estuaries are often used as transportation routes, it is necessary to accurately estimate the amount of dredging required to maintain navigable depths in these water bodies. A significant portion of the pollution load in a water body is typically transported sorbed to cohesive sediments. Therefore, the importance of considering the transport of these sediments in predicting the disposition of pollutants introduced into an estuary cannot be over emphasized. A two-dimensional, depth-averaged, finite element cohesive sediment transport model, CSTM-H, developed during this study may be used as a tool in the field of estuarial management to predict the fate of sorbed pollutants and the frequency and quantity of dredging required to maintain navigable depths. CSTM-H is a time varying model that is capable of predicting the horizontal and temporal variations of the depth-averaged suspended sediment concentrations and bed surface elevations in an estuary, coastal waterway or river. The two- 262 dimensional, depth-averaged advection-dispersion equation with appropriate source/sink terms is solved at each time-step for the nodal concentrations. Previous models are not as comprehensive as they use mathematical descriptions (or algorithms) of the transport processes (that are considered), that are based on limited studies conducted prior to the early 1970 's. In this study field evidence and the considerable amount of experimental research that has been conducted on the mechanics of cohesive sediment transport since that time have been used to develop new algorithms which describe the processes of erosion, dispersion, settling, deposition, bed formation and consolidation. This has resulted in a model whose oredictive capability is improved over that of previous models. A summary of the algorithms and the improvements achieved in the mathematical representations of these transport processes is given next. Deposited estuarial sediments occur in three different stages of consolidation: unconsolidated, partially consolidated and settled (fully consolidated). Unconsolidated deposits, referred to as stationary suspensions, possess a very high water content and low shear strength and are redispersed, or mass eroded, when subjected to an excess bed shear stress. Partially consolidated deposits have a somewhat lower water content and higher shear strength and are resuspended aggregate by aggregate, i.e. undergo resuspension, when subjected to an excess shear stress. Settled, or fully consolidated beds possess a much lower water content, a much higher shear strength and as well are resuspended aggregate by aggregate when subjected to an excess shear. The shear strength and the density of partially consolidated beds have been shown by laboratory tests to increase with 263 depth below the water-bed interface, and as such are vertically stratified. Both stationary suspensions and partially consolidated beds undergo consolidation due to overburden pressure, with the bed density and shear strength increasing with time of consolidation. In settled beds, the shear strength and the density profiles exhibit relatively uniform properties over the depth. The sediment bed schematization incorporated in CSTM-H allows for the above-mentioned three bed sections, and divides each section into a characteristic number of layers. Within each layer, the bed shear strength and density are assumed to vary in a linear manner with depth. The number of layers as well as the shear strength and density profiles in each section must be determined from laboratory erosion tests. Even though a stationary suspension is not a true bed, it is represented as such in order to account for the sediment mass which forms this suspension. Previous models use a constant bed shear strength and bed density for each layer, and use only a single layer for the partially consolidated bed section. Therefore, the stratified nature of partially consolidated beds is not represented in these models. The bed formation algorithm incorporated in CSTM-H uses the assumed linear bed density profile in each layer to iteratively solve for the thickness of bed formed by the deposition of a given mass of sediment. The bed structure (i.e. bed shear strength and density profiles) of the existing bed is adjusted to account for the added sediment mass. Previous models use the assumed constant bed density value in each layer to solve explicitly for the bed thickness. The erosion algorithm simulates the redispersion of stationary suspensions by instantly redispersing the thickness of the bed above the 254 level at which the bed shear stress, -z^, is equal to the bed shear strength, x^. The average resuspension rate of the partially consolidated bed layers over one time-step At is given by an empirical law (Eq. 3.13) that is analogous to the rate expression which results from a heuristic interpretation of the rate process theory of chemical reactions. This rate expression indicates that the resuspension rate varies exponentially with the excess bed shear stress. The average resuspension rate of the settled bed layers is given by an empirically determined expression (Eq. 3.12) that is equal to the first term of a Taylor series expansion of the empirical resuspension rate law for partially consolidated bed layers. Thus, the rate of erosion of settled beds is linearly proportional to the excess shear. The thickness of the partially consolidated bed section or the settled bed section eroded per time-step is determined using an iteration routine. The effect of salinity on the bed shear strength, and hence on the erosion rate of that bed, as determined from laboratory resuspension tests is incorporated into the erosion algorithm. For a natural mud, the bed shear strength was found to double in value, in a linear manner, between S = 0 and 2 ppt, and thereafter (for S > 2 ppt) was found to remain practically constant. Based on an interpretation of typically observed Eulerian concentration-time records in estuaries, erosion is simulated to occur only during temporally accelerating flows when t,^ is greater than T^. Previous models: 1) assume that the erosion rate of both partially consolidated and settled beds varies linearly with the excess bed shear stress, 2) do not account for the effect of salinity on the bed shear strength, 3) assume that erosion occurs whenever, in either a temporally decelerating or accelerating flow, is greater than and 265 4) do not account for the bed shear strength and density profile in each layer (since constant values for and p are used) in determining the mass of sediment eroded. The dispersion algorithm developed in this study utilizes the Reynold's analogy between mass and momentum transfer and solves for the four components of the two-dimensional sediment dispersivity tensor using the formulation derived by Fischer (1978) for bounded shear flows. Thus, only shear flow dispersion is accounted for in this algorithm. The limitations of such a dispersion algorithm are consistent with those associated with a two-dimensional, depth-averaged cohesive sediment transport model. Previous cohesive sediment transport models: 1) do not include the cross product dispersion coefficients in the advection-dispersion equation and 2) do not include a dispersion algorithm to calculate the dispersion coefficients as functions of the local depth of flow and the depth-averaged velocity components. The settling velocity of cohesive sediments is a function of, among other parameters, the suspension concentration, C, the salinity, S, and ■z^. For concentrations less than C^^ = 0.1 - 0.7 g/1 the sediment particles settle independently without much mutual interference, and therefore the settling velocity is independent of C. In the range < C < 10 - 15 g/1, the settling rate is proportional to C" with n > 0, due to mutual interference. In the range C > the settling velocity decreases with increasing concentration due to hindered settling. The deposition algorithm integrates the concepts proposed by various investigators and represents a unified model of this process. Deposition is predicted to occur only in decelerating flows, i.e. 266 tjj(t+At) < "^bCt), when is less than the maximum shear stress at which deposition can occur, \ . For \ < \ r Change I), where - is the '-'max u u,^ u,L value of '^^ at which the deposition rate in Range I is equal to that in Range IIB (defined below), and for C < Ci for all values of \ < \ , "max the rate of deposition is determined using the exponential law given by Eq. 3.28. For \ r ^ \ ^ \ ^nd C > C-, (Range IIB), the deposition "»'- "max ^ rate is given by a log-normal expression (Eq. 3.45). The thickness of the bed formed by a given deposited sediment mass is determined using the properties of the unconsolidated and partially consolidated bed sections. As deposition continues, first the unconsolidated layers are filled up, followed by the partially consolidated layers. Increasing the salinity of the suspending fluid was found to increase slightly the settling velocities, and hence the deposition rates of a natural mud. The settling velocity, W^, was found to increase as s'^*^-^ in Range I. The effect of salinity on the deposition rate in Range IIB is given by Eqs. 3.52 and 3.54, and was approximately the same as that in Ranges I and IIA. The settling/deposition algorithm in previous models: 1) do not include the effect of salinity on the rates of deposition and 2) predict that deposition occurs only when t^k ^ "^h » either temporally min accelerating or decelerating flows. Thus, deposition is predicted to occur in the previous models during only a small percentage (e.g. 20% for kaolinite in tap water) of the shear stress range in which deposition has been observed to occur in laboratory steady flow experiments. The consolidation algorithm accounts for the consolidation of a stationary suspension and partially consolidated bed by increasing the 267 bed density and bed shear strength with time. Consolidation is considered to begin after the bed formation process is complete, at which time the bed thickness will be maximum. After two hours of consolidation the stationary suspension layer(s) become part of the partially consolidated bed, and therefore would undergo resuspension if subjected to an excess shear stress. The variation of the mean bed density with consolidation time is given by Eq. 3.67, while the density profile p{zb) is determined using power law relationships between p and (Eqs. 3.70-3.72). The thickness of the bed is reduced to account for the expulsion of pore water during consolidation, and to insure that the mass of sediment in the bed is conserved. The new deposit bed section of the layered bed model is further divided into a finite number of strata in order to account for repeated periods of deposition, as typically occur in estuaries due to the oscillating tidal flow. The top stratum may be composed of a stationary suspension and partially consolidated bed, whereas the buried strata are composed of just partially consolidated sections. The degree of consolidation of a particular stratum is accounted for by using a separate consolidation time for each stratum. Due to the extremely limited information on bed shear strength profiles in cohesive sediment beds, the variation of ^(zk) with consolidation time is determine using a power relationship between p and -c^, as given by Eq. 3.61. None of the previous cohesive sediment transport models account for the increase in the bed shear strength and density profiles due to consolidation, and thus are not capable of simulating the decrease in the susceptibility to erosion of a consolidating bed with time due to the continual increase in the bed shear strength. 268 The Galerkin weighted residual method is used to solve the advection-dispersion equation for the nodal suspended sediment concentrations. An existing finite element formulation was modified to include the two cross product dispersion coefficients. and D . The Aj y X model yields stable and converging solutions. The accuracy of the solution is affected when the Peclet number becomes too large (greater than lo2) or too small (less than 10"^). Verification of CSTM-H was carried out against four erosion- deposition experiments, three of which were performed in an 18.1 m long recirculating flume and the fourth in an 0.76 m mean radius rotating annular flume. Simulation of a deposition experiment in an 100 m long non-recirculating flume at the Waterways Experiment Station, Vicksburg, Mississippi yielded satisfactory agreement between the measured and predicted deposit thickness in the downstream 65 m reach of the flume. The recorded differences in the upstream 35 m reach may have been caused by the occurrence of longitudinal sorting, or the increased turbulence present in this section of the flume. CSTM-H is not capable of simulating longitudinal sorting because the parameters which characterize the log-normal depositional law, used for deposition Range IIB, are assumed to be spatially invariant. The model was applied to prototype conditions. Sedimentation in Camachee Cove Marina, located adjacent to the Intercoastal Waterway in St. Augustine. Florida was modeled. A mean shoaling rate of 8.6 cm/year, which is representative of fair weather conditions only, was predicted. Sediment transport in a 10 km hypothetical canal, in which both erosion and deposition occurred, was also simulated at four different salinities to show the effect of salinity on the rate of 269 sediment transport under typical prototype conditions. These three simulations demonstrated the significant influence of salinity on the rate of erosion of a cohesive sediment bed for salinities less than 10 ppt, and the reduced effect for salinities greater than about 10 ppt. 6.2. Recommendations for Future Research Based on the conclusions from the present study, the following objectives and recommendations for further research are made: 1. Conduct a thorough laboratory investigation of the consolidation characteristics of cohesive soils. Objectives for such a research program could be the following: 1) verify and/or modify the multi -strata bed formation-consolidation algorithm developed during this study. 2) Determine the validity and practicality of using one of the existing explicit finite difference finite strain consolidation models to predict the consolidation of cohesive soils. 3) Investigate further the possible correlation between the density and shear strength of cohesive soils. 2. A laboratory investigation of the redispersion characteristics of stationary suspensions and partially consolidated beds is recommended. This recommendation is motivated by the differences obtained between the measured and predicted suspension concentrations during the first one-half hour of periods of erosion in the three experiments conducted in the 18.1 m recirculating flume. Erosion experiments using flow deposited cohesive sediment beds should be conducted in which the consolidation time and applied bed shear stress are systematically varied. 270 3. The results from the model simulation of the tidal cycle experiment in the rotating annular flume reveal the need for studying the variation of the vertical suspended sediment profile in an unsteady flow field. It would be advantageous to conduct such an experiment in a rotating annular flume because longitudinal velocity and suspended sediment gradients do not occur in this facility. 4. The deposition algorithm developed during the present study can be modified to include the effect of longitudinal sorting on the rates of deposition by incorporating the variation of tgg and cr^ in the two horizontal dimensions. Extensive field and laboratory tests would be required to determine the following relationships: t^Q = tgg (x.y.x^) and 0^ = {x,y,x^). APPENDIX A DERIVATION OF ADVECTION-DISPERSION EQUATION In a diffusing mixture such as the sediment-water binary system, the various constituents move at different velocities. For example, the vertical advective velocity of the water differs from that of the negatively buoyant sediment particles by the sediment setting velocity, Wg. The local mass averaged hydrodynamic velocity for a binary system is defined as that which would be measured by a pi tot tube: ^ total momentum C V.. + C.V, (A.l) Y - WW s s mass of mixture C + C w s Where = velocity of water mass, V = velocity of sediment mass, C 3 ' w mass of water/total volume of mixture and = mass of sediment/total volume of mixture. Using the coordinate system defined in Fig. 3.1, Y and are defined as \ = ui + vj + wk (A. 2) = ui + vj + (w+W^)k (A. 3) where u, v and w are the fluid velocity components in the respective Cartesian coordinate directions. Here the water and sediment particles are assumed to be advected in the x- and y- directions at the same respective velocity components. 271 272 The advective mass flux of sediment is defined as - %\ (A.4) while the diffusive mass flux is given by J„ = C (V -V) s - S^'s"^' (A. 5) The diffusive flux in Eq. A. 5 is that due to molecular diffusion, which by Pick's first law is equal to 's- V(-) (A.6) P where p = density of binary system and d^^ = molecular diffusivity of the sediment particles in water. In general D^^^ is a function of Cj, C^, ^, T and p, with the latter three parameters being the absolute viscosity, temperature and pressure of the binary system. The total flux of sediment, N3, is equal to the sum of the advective and diffusive fluxes, given by Eqs. A.4 and A.6, respectively. ^s = = ^s^ - PDsw'(-) (A.7) P The law of conservation of sediment leads to the following continuity equation for suspended sediment: ac -^+^.H =S (A.8) at ^ 273 where S = source/sink term to account for the mass of sediment addded or removed per unit volume per unit time. Substituting Eq. A. 7 into Eq. A. 8 gives ac * C ~^''^s\= ^•P'^sw'^-^ ^ S (A.9) ot p In tensor notion Eq. A-9 becomes ac a a a c St ax. ^ ^- ax. ^^ax. p ' J In turbulent flow the instantaneous velocity components and suspended sediment concentration can be expressed as the sum of a time-averaged term and a fluctuating component, i.e. V, = + v' s. s^ s. " ^s S (A. 11) where the overbar and the superscript prime denote the mean and fluctuating quantities, respectively. The mean term is averaged over a time interval, Tj, which is small compared with the time scale for the mean flow, but large compared with the time scale for the turbulent fluctuations so that the time averages of v' and c' over Tt are ^ -J 5 1 approximately zero (MacArthur, 1979). Substituting Eq. A. 11 into Eq. A. 10 and averaging the entire equation over Tt gives ac a - - a , . a a c — + (C V ) + (C Y ) = [pD (-)] + S (A. 12) St sx. ^- ax ax^ swg _ p 274 where the subscript s on the sediment concentration C3 has been dropped for convenience, and where the following definitions have been used: - 1 '''l 1 ^^"^I . J C dt c = — / C dt = 0 (A. 13) Tj t Tj t 1 t+T T t +T V = — / V dt V = — / V dt = 0 i Tj t ^- Tj t ^ The terms C V^. in Eq. A. 12 represent the turbulent diffusive mass transport of sediment due to the turbulent velocity fluctuations in the x^. direction. Reynolds analogy which is based upon the analogy between the transfer of mass and momentum in turbulent flow and upon Boussinesq's eddy viscosity hypothesis is used to relate these diffusive sediment transport terms to the spatial gradient of the time-averaged concentration as follows: I . 5C C V = -E /, ,.v s. X. ^ (A. 14 1 1 Qx. where E^. = turbulent diffusion coefficients of sediment in the x- direction. Substituting Eq. A. 14 into Eq. A. 12 gives ac a a a c a 9r Z^Z'^'^^'^i^ ^r-^P'^swr-^-)^ ^— (E, — ) + S (A. 15) °t ex. 1 dx ax. p ax. ^i Sx. where the bars over the mean parameters have been dropped for convenience. 275 Next, the fraction of sediment mass, C/p, 1n the molecular diffusion term in Eq. A. 15 is simplified by effectively assuming that the mixture density or total mass concentration is not a function of position in the binary system. The physical justification for this assumption is that the effect of spatial gradients of p is included in the molecular diffusivity, D^^. As a result, Eq. A. 15 may be rewritten as 9C a 8 5C — + — (CV ) = (E ) + S (A. 16) at ax. ^• dx. ^ax^. , where E^-j = D^^ + E^^^ is the turbulent diffusion tensor, in which the off-diagonal terms of . are not neglected. Equation A. 16 is the three-dimensional form of the advection- dispersion equation for suspended sediment transport in a fully developed turbulent flow field. Upon expansion of the tensor terms. Eq. A. 16 becomes ac a a 5 a ac — + — (Cu) + — (Cv) + — {C(w+W )} = — (E — + at ax dy 92 ^ ax ^^ax ac ac a 5c ac sc a ac 5c 5c — (E_ — + E^ — + E — ) + S az ^^ax ^^dy ^^dz The desired two-dimensional form of Eq. A. 17 is obtained by integrating this equation over the local flow depth d, which is defined as 276 d = h(x,y,t) - b(x,y,t) (A. 18) where h(x,y,t) and b(x,y,t) = elevations of the water surface and bed, respectively, with respect to a tidal datum. When vertical integration from b(x,y,t) to h(x,y,t) is performed, vertical profiles of the time- averaged velocity components in the x and y directions and the suspended sediment concentration give rise to dispersion terms. In order to account for such terms, the velocity components, u, v and w, and the concentration, C, are expressed as the sum of a depth-averaged term and a term which is the deviation of the parameter over the depth of flow about the depth-averaged value, e.g. ~ II V = V + V (A. 19) C = C + c" where the double bar and the double prime denote the depth-averaged and the deviating quantities, respectively. The following definitions of the depth-averaged and deviating terms are used: = 1 h h e = - J e dz with / e dz = o (A. 20) d b b for 9 = u,v,w, and C. Equation A. 19 is substituted into Eq. A. 17 and the entire equation is integrated from b{x,y,t) to h(x,y,t) using Leibnitz rule. The result is given on a term-by-term basis below: u + u w = w + w 277 h ac h a = „ / _dz = / — (C+c )dz b at b at ab c"(b,t)— at a = = a(h-b) — {C(h-b)} - c at at ah c (h,t) — + at (A. 21) ha h a = „ = „ a J — (Cu)dz = / — {(C+C ){u+u )}dz = — {Cu(h-b)} - b ax b ax 5x ..= ah = ah .. ah {Cu}| {c ull — - {cu }| — - {c u }| — + h ax h ax h ax h ax a h „ „ — / C u dz ax b (A. 22) ha a == == 5h ,.= dh J — (Cv)dz = — {Cv(h-b)} - {Cv}l {C v}| b ay dy h ay h ay ah „ „ 5h a h „ „ {Cv }| {c V }| — + — / c V dz (A. 23) h ay h ay dy b ha =.. = .. J — (Cw)dz = {(c+c )(w-H^ )}| (A 24) b az h h a / — (CW Jdz = {CW }| - {CW }! b az 5 2 h ^ b ha ac a ac ac a / r^^xx"^^^ - — {{h-b)E — } - (E^ — )— (h-b) - b ax ^^dx ax ^^ax ^^ax ax >■ II ac ah dc ab a „ sh {E ■ }| _+{E }| {E [c] — - ''''ax h ax ''''ax ^ ax ax ^ ax „ ab t^c \ — )} (A. 25) ax 278 J — (Evv— = — {(h-b)E, — } - (E, — )— (h-b) - b ay y^dy ay yy ay yy ay ay ac ah ac ab a „ ah y^ay h ay yyay b ay ay yy ^ Sy „ ab ay (A. 26) h a ac ac 5c / — )dz = {E^^ }| - {E }| b az az ^^az h az b (A. 27) h a dc a ac ac a ^ r^^xv"^'^^ = — {(h-b)E — } - (E — )— (h-b) - b ax ^yay ax ^yay ^y ay ax ac ah ac ab a ah ay h ay xy ay b ax ax xy 'ay ab ay (A. 28) h a ac a ac ac a J — )dz = — {(h-b)E^— } - (E — )— (h-b) b ay ^yax ay xy ax xy ax ay ac ah ac ab „ ah ^yax h ay ^yax bay ^y "ax „ Bb Cc 1— )} ''ax (A. 29) ha ac a h ac / — (E _)dz = — V ax ^^az / E ax b ^^az ac - {E }| + XZa u az h 279 9c (E — }l OZ b (A. 30) ha ac a h ac / — (E^ ~)dz = — / E — b Sy y^az dy b y^az II ac y^az b ac - {E — }| + y^az h (A. 31) h a 5c a = „ 5 = / — )dz = — (C+c )]| - {E — (C+c")}i h az '^^ax ^^ax h ^^ax b (A. 32) ha dc 5 = 5 = ay (A. 33) Substituting Eqs. A. 21 through A. 33 into Eq. A. 17 and dropping the double overbars from the depth-averaged terms gives a a a ahnu ah — (dC) + — (dCu) + — (dCv) + — / u"c" dz + — J v"c"dz= dy ax b dy b a ac ac a 6c sc ■^^^yy + } + {dE — + dE — } + ax ^^ax >^yay ay ^^dx y^ay " " a h ac a h ac — / dz + — / E dz + ax b ^^az Qy b y^ay h H + B + / Sdz (A.3A) b 280 where H represents the sum of the outward normal flux of sediment and change in storage at h(x,y,t), and B represents the sum of the normal sediment flux out and storage change at b(x,y,t). The expressions for H and B are given below: [C+c ] — + [(c+c ){u+u [(C+c )(v+v )]^— "at ^x ■'dy II II ah [(c+c )(w+w )\ - C(C+c )W 1. - E [—(C+c )]^ s n XX ax 'ax ah ac ah Eyy[— (C+c )],— + [E„ ], - E_[(C+c )],— - ay ay ^^az ^ 'ax ah ac ac E^y[— (C+c )\~- - [E.,_ ] ax ■ay ^1. a „ a „ dh [E — (C+c )] + [E — (C+c )] - — {E„Jc + x^5x " y^ay ax ^ax I, ah „ ah ay ,1 ah ax' (A. 35) II ab „ a „ ab B = -[C^-C [(Cc )W^], . E^/-(C*c )]^-. ax ax 'at a 11 ab ac a „ ab EyyMc+c \—- [E — r + E [— (C+c )].— + ay °ay a „ ab ac ac ax ^ay " ^-^^^az ' ^^y^az'b ° .1 a 5 „ ab [E.,— (C+c )] - [E —(C+c )] + — {E [c ]— + xz ax ay ^ ax ^x 281 I. 9 „ Sb 9b The terms H and B represent the boundary conditions at h(x,y,t) and b{x,y,t), respectively, since Eq. A. 17 was vertically integrated from h to b. H is equal to zero since it is assumed that there is no net rate of transport of sediment across the instantaneous free water surface. Therefore, Eq. A. 34 can be simplified to yield 0 0 9 11 ■ II 9 — (dC) + u — (dc) + V — (dC) + — (du"c") + — {dv"c") = at 9x 9y dx 9y a 9c 9c 9 ac ac ""^'^^yy" ^^vv— ^ + — ^dE ~ + dE — } + ax ^^ax ^y9y 9y y>^ax yyay II a h 9c 9 h ac h Z I ^z7-^' r / ^z7-^^ ^ ^ ^ Sdz (A. 37) 9x b 9z 9y b b where the double overbar denotes the depth-averaged value of the quantity thereunder, and where the assumption of an incompressible fluid (i.e. V.u = 0) has been utilized. A Reynolds analogy is again used to relate the quantities under the double overbars in Eq. A. 37 to the spatial gradient of the depth- averaged suspension concentration as follows: 11 II 9c u.c = -K. . — 1 IJ; 'ax, 1 .J 1,2 (A. 38) 282 where K^-j = two-dimensional sediment dispersion tensor. Substituting Eq. A. 38 into Eq. A. 37 gives a 5 a a 5c 5c — (dC) + u— (dC) + V— (dC) = — {dD — + dD — } + at ax ay ax ^^ax ^^ay a ac ac a h ac — {dD — + dD — } + — / E^^ dz + ay y^ax y^ay dx b ^^^z II a h ac h — / E — dz + B + / Sdz (A. 39) ay b y^az b where D^-j = K^.j + Z. ■ = effective sediment dispersion tensor. If Ey2 and Ey^ are both assumed to be linear functions of z such that the partial derivatives of E^^ and Ey^ with respect to z are functions of only x and y, respectively, the two integrals in Eq. A. 39 become a h ac a ax b ^^^z ax X2 h xz b (A. 40) s h ac a (A. 41) The first and second terms on the right hand sides of Eqs. A. 40 and A. 41 should be incorporated into H and B, respectively, as they represent fluxes of sediment out of the water surf act and bottom. 283 Therefore, Eq. A. 39 becomes 9 5 9 5 5C dC — (dC) + u— (dC) + V— (dC) = — {dD — + dD — } + at 9x 5y 5x ^^5x ^^ay B ac ac where S = depth-averaged source/sink term. The term B in Eq. A. 42 represents the rate of erosion and deposition at the bed-fluid interface, b(x,y,t). That is, sediment that settles or diffuses out of suspension is part of the depositional flux, and sediment that advects or diffuses into suspension is part of the erosional flux. Therefore, B can be alternatively expressed as d d B = e + Ti = — (dC)| +— (dC)| (A. 43) dt Erosive dt Depositional Flux Flux in which s = idC)] _ = dry mass of sediment eroded per unit dt Erosive Flux bed surface area, and ti = MdC) |^ ^^^^ ^ ^^^^ sediment dt deposited per unit time per unit bed surface area. The depth-averaged source/sink term S represents the rate of sediment influx to or outflux from the water body not attributable to either the processes of erosion or deposition. For example, S would account for the removal (sink) of a certain mass of sediment by dredging in one area (e.g. navigational channel) of a water body, and the dumping (source) of the sediment as dredge spoil in another location in the same body of water. 284 So Eq. A. 42 is the two-dimensional, depth-averaged advection- dispersion equation which governs the transport, addition (i.e. source) and removal (i.e. sink) of suspended sediment in a turbulent flow field. Expressions for £ and ti are given in Sections 3.4 and 3.6, respectively. The total dispersive fluxes, f, in the x- and y- directions are given by ac ac f„ = dD — + dD — (A. 44) X xxg^ xy^y ^'^•'^^^ ac ac f = dD — + dD — (A 4"^^ y yxa^ yy^^ ^'^•^s) which are the expressions inside the curly brackets in Eq. A. 42. Using the coordinate system shown in Fig. 3.1, the dispersive fluxes in the longitudinal (along the flow axis) and transverse (perpendicular to flow axis) directions, f-, and f^, respectively, are related to f, and f„ as ' X y follows: -f^cos Q - f^sin e fy = f^sin 9 + f^cos e (A. 46) (A. 47) where f-, and f^ are given by ac f, = -dD, — (A. 48) ac f^ = -dD — (A. 49) an 285 in which © = arctan (u/v), D-j = effective longitudinal dispersion coefficient and = effective transverse dispersion coefficient. The four components of the two-dimensional dispersion tensor, D^-j, are related to D-] and by the following functional relationships: D^^ = D^cos^e + D^sin^e (A. 50) D^y = =V^D^-D^)sin(2e) (A. 51) D = D^sin^e + D^cos^e (A. 52) Expressions for D^-j are given in Chapter III, Section 3.5. APPENDIX B COEFFICIENT MATRICES IN THE ELEMENT MATRIX DIFFERENTIAL EQUATION The purpose of this appendix is to list the matrices and vectors in the element matrix differential equation (4.23) and describe how they and the contour integral in Eq. 4.22 are evaluated. The product of the element coefficient matrix, [k], and the nodal concentration vector, {C}^, is seen from Eqs. 4.22 and 4.23 to be equal to A .dc .5c ^'U ac [k]{c}^ = // [N (u— + v— ) + [D — + e ax ay 5x ^^ax ac ac^ dc where the approximate velocities, u and v, are evaluated at each point (C,Ti) as follows: l=n ^ i=n u = 2 N.u. v = 2 N.v. (B.2) 1=1 ^ ^ 1=1 ^ ^ The dispersion coefficients are considered not to vary significantly in either space or time in this formulation, and therefore are assumed to be constants. The transformation from global coordinates to local element coordinates derived in Chapter IV, Section 4.4.2, gives 286 287 dxdy = lJ|5dTi {B.3) Substituting Eqs. B.2 and B.3 into Eq. B.l and dividing through by the concentration vector {C}^ gives the following expression for the (i,j) term of [k]: ' ^ ^ ax ^ ^ay ax ax aN. an . aw. aN . an. aN. a7 ~ ^y'^ a7 aJT "^yy a7 |J|dCdTi for 1=1, n (B.4) The notation 1=1, n indicates that the variables with the subscript 1 are summed from 1=1 to l=n=number of nodes in a particular element. Thus, the element coefficient matrix is given by i=n j=n l=n [k] = 2 Z E k (B.5) i=l j=l 1=1 The double integration in Eq. B.4 is performed using the Gauss- Legendre quadrature formula as follows (Ariathurai et a1_., 1977): . , m=NQ mm=NQ /-I i_i f(^,il) d^dTi = S 2 H H f(| ,Ti ) (B.6) ^ ^ m=l mm=l ™ 288 where f=k Is given by Eq. B.4, NQ= number of quadrature points and H=weight factors. The element temporal matrix [t]^ is given by i=n j=n l=n Itl = >: Z Z // N H N d, [J I d^dn (B.7) i=l j=l 1=1 % 1 J ' I where again the integration is performed using the Gauss-Legendre method. e The element source/sink vector {f} is given by i=n j=n l=n if} = - ^ Z Z // N-N^dTSlJldldn (B.8) i=l j=l 1=1 e ^ ' ' and the boundary matrix [b]^ is given by i=n j=n l=n ' j- 2 2 E / M N d, [(D + D^^ ) n^ i=l j=l 1=1 ^ 1 ' XX 5x ay X The boundary matrix [b]e accounts for a specified concentration flux boundary condition along the boundary of a domain boundary element. APPENDIX C COMPUTER PROGRAM The computer program of CSTM-H is written in FORTRAN IV using double precision arithmetic. Double precision is required in simulating laboratory scale tests in order to minimize round-off error. Prototype systems can be modeled using single precision arithmetic. A description of the main program and the subroutines followed by a flow chart and user's manual are presented. C.l. Main Program In the main program the following information is read: input/output file numbers, problem option and control parameters, finite element grid geometry and transient control parameters. The I/O file numbers determine the file numbers used in both reading and printing certain data. The problem option parameter, NOPT, specifies which one of four types of problem is to be solved. The four types of problems that can be solved by CSTM-H are: 1) steady state advection-dispersion of a conservative constituent, 2) unsteady advection-dispersion of a conservative constituent, 3) cohesive sediment transport and 4) one- dimensional consolidation of a clayey sediment bed. The flow chart in Section C.3 diagrams the program steps used in the solution of each problem type. The transient input code arrays specify at which time steps new boundary conditions, flow depths, velocity field, dispersion 289 290 coefficients, sediment settling velocities, density (i.e. salinity) field or local source/sink terms (explained in Appendix A) are read in or calculated using either built-in algorithms or user supplied routines. The user's manual in Section C.4 gives a short definition of each parameter read in either the main program or one of the subroutines. The grid data that need to be read includes the number of nodes, NP, number of elements, NE, nodes forming each element (nodal connections in counterclockwise direction) and the two horizontal coordinates of each node. The main program also initializes necessary arrays, prints out the initial conditions and initial values of certain parameters, contains the main time loop and prints out the concentration-time record for specified elements. C.2. Subroutines A brief description of the subroutines and subf unctions in CSTM-H is given below. Subroutine BAND - Forms and solves the system matrix equation at each time step for the nodal concentrations using Gaussian elimination. This subroutine is used when ISOLV = 0. Subroutine BEDFOR - Forms the sediment bed that is a result of 1) deposition during the previous time step or 2) new deposits present on top of the settled bed at the start of the modeling. Subroutine BEDMOD - Control program for bed formation and consolidation routines. Determines mass of stationary suspension for which T^^^, = 2 hours; calls Subroutine BEDFOR, and stationary suspension becomes part of the partially consolidated bed. Determines the consolidation time 291 for each bed stratum, and calls Subroutine CONSOL for the strata with Tjj, > 2 hours. Subroutine BEDSS - Computes the bed shear stress at each node. Calculates the average flow depth, velocity, bed shear stress, water density and concentration for each element using the element shape functions. Compares this time step's bed shear with the previous time step's bed shear to determine if the flow at each node is temporally steady, accelerating or decelerating. Subroutine COMPAR - Compares analytical solution with numerical solution for steady state problems. Subroutine CONCBC - Reads or computes concentration boundary conditions at specified nodes. Subroutine CONCIC - Reads or computes the initial suspended sediment (for sediment transport problem) concentration at every node. Subroutine CONSOL - Computes the increase in the bed density profile in each stratum due to consolidation. The bed strength profile is computed as a function of the new bed density profile. Subroutine DENSITY - Computes the water density at every node using the given water temperature, salinity and suspension concentration. The kinematic viscosity is also calculated as a function of the water temperature. Subroutine DEPMAS - Computes the dry sediment mass deposited during the previous time step for every element where deposition is predicted to occur. Subroutine DEPSN - Computes the rate of deposition at each node where deposition is predicted to occur. 292 Subroutine DEPTH - Reads or computes (using user specified procedure) the depth of flow for each node at the time steps where the appropriate transient code array indicates that a change in depth occurs. Subroutine DISPER - Reads or computes (using the dispersion algorithm) the four components of the two-dimensional dispersion tensor for each node at every time step where the appropriate transient code array indicates that the values of the dispersion coefficients change. Subroutine DRYNOD - Determines at which nodes and elements the water depth is negative (i.e. dry). These nodes are eliminated from the system array coefficient matrices. Subroutine ELSTIF - Forms the element coefficient and load matrices. Modifies element load matrix to account for specified boundary conditions. Subroutine EXACT - Computes analytical solution to steady state convection-diffusion problem. Subroutine FRONT - Forms and solves the system matrix equation at each time step for the nodal concentrations using the frontal elimination routine. This subroutine is used when ISOLV = 1. Subroutine ITERC - Computes thickness of partially consolidated bed formed by specified dry mass of sediment using an iteration procedure. Called by Subroutine BEDFOR. Subroutine ITERM - Computes thickness of stationary suspension formed by specified dry mass of sediment using an iteration procedure. Called by Subroutine BEDFOR. Subroutine LOAD - Forms the array NBC which numbers the equations in the system matrix. Number of equations is equal to the number of nodes minus the number of nodes at which boundary conditions are specified. 293 Computes the band width for the system coefficient matrix. This subroutine is called when Subroutine BAND is used to solve the system matrix. Subroutine LOADX - Forms the array NBC. This subroutine is called when Subroutine FRONT is used to solve the system matrix. Subroutine ORGBED - Reads the original settled bed profile and the initial dry mass per unit bed surface area of new deposits on top of the settled bed for elements where such exists. Subroutine RECORD - Records the values at each time step of various parameters for the elements where time records are desired. Subroutine RED - Called by Subroutine FRONT to read data from temporary disc data file. Subroutine REDISP - Computes the redispersion rate for unconsolidated new deposit layers (stationary suspension) when the flow is accelerating and the bed shear stress is greater than the shear strength of the suspension surface. Computations are made on an element-by-element basi s. Subroutine RESUSP - Computes the resuspension rate for exposed partially consolidated bed layers or settled bed layers when the flow is accelerating and the bed shear stress is greater than the shear strength of the bed surface. Computations are made on an element-by-element basi s. Subroutine SEDPRP - Reads the settling velocity, new deposit and consolidation properties of the cohesive sediment. Settling velocity parameters and the new deposit properties are printed out. Subroutine SETVEL - Reads or computes using a built-in algorithm the sediment settling velocity in Range I and for C < Ci in Range II as a 294 function of the suspension concentration and salinity at each node for the time steps where the appropriate transient code array indicates that a change in the settling velocity occurs. Subroutine SHPFNS - Computes the isoparametric quadratic shape functions and their derivatives for quadrilateral elements with parabolic sides. Subroutine TSHAPE - Computes the isoparametric quadratic shape functions and derivatives for triangular elements with parabolic sides. Subroutine VEL - Reads or computes using a user specified routine the depth-averaged components of the velocity in the x- and y- directions at each node for the time steps where the appropriate transient code array indicates that a change in the velocity occurs. Subroutine WRITER - Prints out the bed shear stress, bed elevation, erosion/deposition rates and the layer-by-layer bed properties for each element at each time step where the appropriate transient code array specifies. Subroutine WRT - Called by Subroutine FRONT to perform mass transfer of data to temporary disc data file. Function DENFUN - Computes the water density as a function of temperature and salinity. Function FBETA - Computes the value of the empirical coefficient A in Eq. 3.68 as a function of T^^.. Function FDELTA - Computes the value of the empirical coefficient B in Eq. 3.68 as a function of J^^. Function FMBDN - Computes the value of the final mean bed density (given by eq. 3.64) as a function of Cq. Function FTCIN - Computes the value of T^^^.^^ (given by Eq. 3.65) as a function of C^. 295 Function FZHMIN - computes the value of z^^^^ (given by 3.69 with 9 = z^^^) as a function of T^^,. Function SIGFUN - Computes the value of 02 (given in Eq. 3.36) as a function of t;|^. Function T50FUN - Computes the value of tgg (given in Eq. 3.36) as a function of Tj^. C.3. Flow Chart 296 START J READ I/O FILE NUMBERS COORDINATES NO 297 0 1 / READ TRANSIENT INPUT DATA READ ELEMENT NUMBERS FOR WHICH TIME HISTORY IS TO BE PRINTED OUT INITIALIZE NECESSARY ARRAYS READ AVERAGE WATER TEMP. AND INITIAL SALINITIES READ SOURCE/SINK TERf-i AT APPROPRIATE NODES YES SEDIMENT PROBLEM - INITIALIZE BED PROPERTIES 5 6 299 i CALL DENSTY- ELEMENTAL BED SHEAR STRENGTHS AND BULK DENSITIES CALCULATED AS A FUNCTION OF ' ELEMENTAL SALINITY VALUES 300 CALL VEL- READ INITIAL VALUES OF FLOW VELOCITIES CALL BEDSS- CALCULATE NODAL BED SHEAR STRESSES CALL DISPER- READ/CALCULATE INITIAL DISPERSION COEFFICIENTS CALL SETVEL- YES READ/CALCULATE NODAL SETTLING VELOCITIES 301 CALL DRYNOD- DETERMINES WHICH NODES AND ELEMENTS ARE DRY CALL LOAD- NUMBER OF EQUATIONS IN THE SYSTEM COEF. MATRIX AND BANDWIDTH DETERMINED CALL LOADX- NUMBER OF EQUATIONS IN THE SYSTEM COEF. MATRIX AND BANDWIDTH DETERMINED PRINT INITIAL CONDITIONS AND SEDIMENT PROPERTIES STEADY-STATE PROBLEM YES YES NO NO UNSTEADY AND SEDIMENT PROBLEflS- NO YES 18 302 CALL FRONT- NODE SOURCE TERMS ADDED, GLOBAL COEF. MATRIX FORMED AND SOLVED BY FRONTAL ELIMINATION ROUTINE USING FULL PIVOTING YES CALL BAND- NODE SOURCE TERMS ARE ADDED INTO SYSTEM LOAD MATRIX, GLOBAL COEF. MATRIX FORMED AND IS SOLVED BY GAUSSIAN ELIMINATION CALL DEPSN- DEPOSITION RATES CALCULATED CALL RESUSP- SURFACE EROSION RATES CALCULATED NO CALL WRITER - PRINT CONCS. CALL ELSTIF- ELEMENT STIFFNESS ARRAYS FOR FIRST TIME STEP FORMED CALL COMPAR- COMPARE WITH EXACT SOL. MAIN TIME LOOP DO N=2,NTTS DEPENDING ON INPUT CODES, READ NEW PARAMETERS FOR THIS TIME STEP CALL DENSTY- SET NEW SALINITIES CALL CONCBC- SET NEW BOUNDARY CONDITIONS CALL DEPTH - SET NEW FLOW DEPTHS YES CALL DISPER- SE! NEW DISPERSION COEFFICIENTS CALL SETVEL- SET NEW SETTLING VELOCITIES READ NEW NODAL SOURCE/ SINK TERMS 305 " CALL DRYNOD CALL 1 -OADX YES CALL DEPSN CALL REDISP- REDISPERSION RATES CALCULATED CALL RESUSP CALL BAND CALL FRONT 306 IF iNOPT EQ NO YES CALL DEPSN- DEPOSITION RATES CALCULATED CALL BEDMOD- CONTROLS BED CONSOLIDATION ALGORITHM CALL RECORD- OUTPUT FOR THIS TIME STEP SAVED II 307 308 C.4. User's Manual SET A CARD A.l 1-5 6-10 11-15 15-20 21-25 26-30 31-35 36-40 41-45 46-50 51-55 56-60 61-65 CARD A. 2 1 2-78 CARD A. 3 1-5 JOB CONTROL CARDS 6-10 11-15 16-20 (1315) IN LP INC IND INE INF ING INH INI INB INS INSS ISOLV (11, 19A4) NSTOP TITLE (1615) NOPT ICODE NTTS IVEL I/O file numbers and equation solver used General input filt number (default 5) Output file number (default 6) Initial concentrations Diffusion coefficients Mode point bottom elevations (initial) Node point flow velocities Settling velocities Flow depths Finite element grid geometry data Boundary conditions Salinities New nodal . salinities 0 - uses band solver 1 - uses frontal solution technique Job stop and title 0 - continue 1 - end of job Job title Job control parameters, input codes, and problem options Type of problem 1 - steady state transport problem 2 - unsteady transport problem 3 - sediment transport problem 4 - consolidation problem only Output control for non-sediment problems 0 - standard output 1 - compares with analytic solution calculated in Subroutine EXACT Number of time steps Determines initial velocity field, i.e., at time step #1 (for unsteady problems only) 1 - velocity components in x and y directions are set equal to constants CONXV and CONYV read in Subroutine VEL 2 - each nodal velocity read in from input file INF 3 - velocity computed using user supplied routine in Subroutine VEL 309 21-25 : lELEV 26-30 : IDIFl 31-35 : IBED 36-40 : ISET 41-45 : IDEP 46-50 : ICONC 51-55 : INBC 46-60 : IDRY Elevation of bottom above a given datum at node points 0 - all elevations set equal to 0.0. 1 - read each value read in from file number INE Initial diffusion coefficient values at each node 1 - and D are set equal to constants read in Subroutine DISPER 2 - nodal diffusion coefficients are read in from file number I NO 3 - diffusion coefficients are calculated using user supplied procedure Initial bed profile 0 - no sediment present on bed 1 - bed profile read in Subroutine ORGBED Initial settling velocity at each node 1 - set to a constant read in Subroutine SETVEL 2 - each settling velocity is read in from file number ING 3 - settling velocities are computed from model in Subroutine SETVEL Initial depts of flow at each node 1 - set to constant read in Subroutine DEPTH 2 - read in from file number INH 3 - computed accordign to user supplied procedure in Subroutine DEPTHS Initial suspended sediment concentrations 1 - set to constant 2 - read in from file number INC 3 - computed according to user supplied procedure in Subroutine CONCIC Boundary conditions 1 - each value read in from file number 1MB 2 - computed in Subroutine CONCBC using user supplied routine Code to indicate dry node (i.e. negative flow depth) problem 0 - no dry nodes will occur 1 - possible dry nodes 51-65 : ISS Code to indicate whether sediment trans- port problem occurs in steady or unsteady flow 310 0 - unsteady flow 1 - steady state flow 66-70 : ISOUR Code to indicate if local sediment source of sink is located at any node 0 - no source/sink 1 - source/sink occurs at one or more nodes SET B MESH DATA These data are read unformatted from file unit INI CARD B.l NE Number of elements in system NP Number of nodes in system CARD B.2 et.seq. NOPd.K) Nodal connections read counterclockwise (8 percard for quadrilateral element, 6 for triangular element) CARD B.3 et.seq. C0RD{J,1) X-coordinate (meters) C0RD(J,2) Y-coordinate (meters) SET C TRANSIENT PROBLEM INPUT CARD C.l TETA DT TIM(l) CARD C.2 et.seq. (8011) l,etc. : NPMA(I) for I=1,.„.,NTTS Transient input Degree of Implicitness for Crank-Nicholson time marching scheme 0 - explicit 1 - implicit Time step - sees (should be of the order 600-1800 seconds for sediment transport problems in estuaries) Starting time - sees Code to change time step The value of NPMA at each time determines if the time step will be changed 0 - no change 1 - double time step 2 - halve time step 311 CARD C.3 et.seq.(80Il) l,etc. : IFFd.l) CARD C.4 et.seq.(80Il) l.etc. : IFF{I,2) CARD C.5 et.seq.(80Il) l.etc„ : IVCOD(I) CARD C.6 et.seq.(80Il) l.etc. : IDIF(I) CARD C.7 et.seq.(80Il) l,etc. : IDEPC(I) Code for new boundary conditions Determines if there are new boundary conditions 0 - no change in boundary conditions 1 - each value read in from cards 2 - computed in Subroutine CONCBC using user supplied procedure 3 - each value read in from file number INB Output control 0 - no output 1 - sedimentation data only 2 - concentrations only 3 - concentrations and sediment transoort data ■ New velocities Same as IVEL but for each time step 0 - no new velocities 1 - X 8 Y velocities set equal to constants CONXV and CONYV read in Subroutine VEL 2 - each nodal velocity read in from input file INF 3 - velocity computed using user supplied routine in Subroutine VEL New dispersion coefficients Same as IDIF 0 - no new dispersion coefficients 1 - D^ and D are set equal to constants read in Subroutine DISPER 2 - nodal dispersion coefficients are read in from file IND 3 - dispersion coefficients are calculated using dispersion algorithm New depths of flow Same as IDEP 0 - no new depths 1 - set to constant read in Subroutine DEPTH 2 - read in from file IHN 3 - computed according to user supplied procedure in Subroutine DEPTH 312 CARD C.8 et.seq.(80Il) l.etc. : ISALC(I) CARD C.9 et.seq.(80Il) l.etc. : ISVS(I) CARD C.IO et.seq.(80Il) l.etc. : ISORS(I) CARD C.ll NHIS NELH Mew salinities 0 - no new salinities 1 - set to a constant read in Subroutine DEM STY 1 - new salinities at specified nodes are read in Subroutine DENSTY 3 - salinities for all nodes are read in Subroutine DENSTY Only for sediment problems H0PT=3 New settling velocities. Same as ISET 0 - no new settling velocities 1 - set to a constant read in Subroutine SETVEL 2 - each settling velocity is read in from file number JNG 3 - settling velocities are computed from model in Subroutine SETVEL Code for local sediment source or sink Same as ISOUR 0 - no source/sink 1 - source/sink occurs at one or more nodes Output control Number of elements for which time history will be written Element numbers For Steady State Sediment Problems (ISS.NE.O), specify at which nodes and elements erosion and deposition initially (i.e. first time step) occur CARD C.12 et.seq.(80Il) Code for nodes l.etc. : ISTP(I) for 1=1,..., NP 0 - deposition occurs initially 1 - erosion occurs initially CARD C.13 et.seq.(80Il) Code for elements l.etc. : ISTE(I) for I=1,...,NE 0 - deposition occurs initially 1 - erosion occurs initially SET D WATER AND SEDIMENT PROPERTIES Read in Subroutine DENSTY 313 CARD D.l (FIO.5,110) Water parameters 1-10 11-20 TMP IS Average water temperature CO Determines how initial salinities are read in 0 - constant salinity for all nodes 1 - salinity for each node is read in Constant salinity Value of constant salinity - ppt If IS = 0: CARD D.2 (F10.5) 1-10 : SW If IS = 1: CARD D.2 et.seq.(7F10.5) Nodal salinity values 1-10 : SAL(I) Salinity value for ll!l node - ppt If ISOUR.NE.O, read source/sink term at appropriate nodes CARD D.3 (4(110, FIO. 5) 1-10 : IT(J) Node number 11-20 : TEMP(J) Local source/sink term - Kg/m^ Reading stops for IT(J) < 0 Read in Subroutine SEDPRP for sediment problems (N0PT=3) CARD D.4 1-10 11-20 21-30 31-40 CARD D.5 1-10 11-20 21-30 31-40 41-50 51-60 CARD D.6 1-10 11-20 (4F10.5) Settling velocity parameters CRCN=C, See equations for below - CRCN2=Cp See equations for below - CRCN3=C^ See equations for below - GAC Density of sediment mineral (6F10.5) AA=Ai See equations for below AB=Ao See equations for ^s below AC=Ao See equations for ^^s below B See equations for < below F See equations for Ws below AL See equations for ^^s below (2E10.3) WSl=W3i See equations for below - Equivalent sediment particle diameter at ^50 ■ 314 CARD D.7 (2F10.5) 1-10 11-20 CARD D.8 1-10 11-20 21-30 CARD 0.9 1-10 11-20 EXPNl=ni EXPN2=n2 (3F10.5) EXPMl=mT EXPM2=nip EXPM3=m3 (2E10.3) WSK1=K WSK2=K; 1 See equations for below See equations for below See equations for W below See equations for W- below See equations for W below See equations for below - m/s See equations for below - m/s NOTE: For RANGE I and C < C^ in Range II = AA*WS1*(SAL)**EXPM1 for C < CRCNl where SAL = salinity If (SAL < 0.1 ppt)SAL = 0.1 ppt = AA*WSK1*C**EXPN1*(SAL)**EXPM1 for CRCNl < C < CRCN2 W3 = AB*WSK2*C**EXPN2*(SAL)**EXPM2 for CRCN2 < C < CRCN3 Wg = AC*G*D**2*(GAC/GAW-1)*250*(C/CRCM3-1)**AL*(SAL)**EXPM3/ (18*v*D**1.8) for C > CRCN3 If there is only one W = kc" relationship between C = CRCNl and the concentration at which hindered settling begins set CRCN3 = CRCN2 in CARD D.4. NOTE: For C > C^ in Range II T=AL0G10((T/T5q)*B*(SAL)**F)**(1./SIG2) CARD D.IO (2I10, 2F10.5) Properties of new deposits I-IO 11-20 21-30 31-40 NLAYTM NLAYT TAUMIN TAUMAX Number of layers formed by unconsolidated new deposits (UND) Number of layers formed by partially consolidated new deposits (CND) mm max Parameters characterizing functional relationship between and log^Q (tgQ) and CT^ at a salinity of 35 ppt. CARD D.ll 1-10 11-20 (5F10.5) Al Sl See equations for 02 below See equations for 02 below 315 A2 See equations for 31-40 : S2 See equations for 41-50 : CI See equations for CARD D.12 (5F10.5) A3 See equations for 11-20 : S3 See equations for 21-30 : A4 See equations for 31-40 : S4 See equations for 41-50 C2 See equations tor NOTE: For CI : a, = Sl*-ug + Al t > CI : ^2 = S2*i;g + A2 C2 : %0 = eo.^io^s^*"^ For 60.*10(S3*i:g C2 : ^50 " J50 •^50 ^50 ^50 So below below below below below CARD D.13 + A4) et.seq.{4(2F10.5)) Shear strength and dry sediment density for unconsolidated new deposit layers. NLAYTM pairs of values are read in starting at the bed surface and proceeding down to the bottom of the bottom UND layer. 1-10 11-20 SSM(I) GADM(I) Bed shear strength - N/m^ Dry sediment density - kg/rn^ for I = 1,..., NLAYTM CARD D.14 et.seq.(F10.5) 1-10 for I CARD D.15 : TLAYM(I) 1,..., NLAYTM Thickness of unconsolidated new deposit layer - m et.seq. (2F10.5) Shear strength and dry sediment density for partially consolidated new deposit layers. NLAYT+1 pairs of values are read starting at the top of these layers and proceeding downward. 1-10 11-20 for I CARD D.16 1-10 11-20 21-30 SS(I) GAD(I) Bed shear strength - N/m^ Dry sediment density - kg/rrr' 1,..., NLAYT+1 et.seq. {3F10. 5) Thickness, and oc values for each partially consolidated new deposit layer. TLAY(I) Layer thickness - m EPSLON(I) - kg/m^/s ALFA(I) a - dimensionless 315 for I = 1,...,NLAYT NOTE: The properties read 1n on CARDS D.13-D.16 are determined from laboratory experiments (see Appendix D, Section D.2 for a description of these experiments). These are the properties assigned to new deposits if/when deposition occurs during model simulation or initially if new deposits are present on top of the original settled bed, as specified in SET K. CARD D.17 1-10 11-20 21-30 CARD D.18 1-10 11-20 21-30 31-40 CARD D.19 1-10 11-20 CARD D.20 1-10 11-20 1-10 11-20 (3F10.5) Empirical coefficients in expression for mean bed density as a function of consolidation time, T^ . ^c AP f in Eq. 3.75 ALAMDA p in Eq. 3.75 TCC Time at which the coefficients A and B in Eq. 3.76 become constants (4F10.5) Empirical coefficients in expressions for final mean bed density and corresponding consolidation time, T^^, . FMBDO (pJq in Eq. 3.72 AI a in Eq. 3.72 TCINO ^^dcjo 1'" ^<^' 3.73 BI b in Eq. 3.73 (2F10.5) Empirical coefficients in expression for A (in Eq. 3.76) given by Eq. 3.77. AQ D (with 9 = A) in Eq. 3.77 BQ F (with 0 = A) in Eq. 3.77 (2F10.5) Empirical coefficients in epression for B (in Eq. 3.76) given by Eq. 3.77. EQ D (with 9 = B) in Eq. 3.77 FQ F (with 9=8) in Eq. 3.77 CARD D.21 (2F10.5) Erppirical coefficients in expression for max given by Eq. 3.77 PQ SQ For NOPT = 2 or 3: D (with e = z.'^g^) in Eq. 3.77 ^ ^""'^^ ® = ' max) 1" Eq. 3.77 SET E INITIAL CONCENTRATION FIELD The initial concentration at each node must be specified for all unsteady problems. The type of input is determined by the value of ICONC. 317 Read from file unit INC ICONC = 1 Initial concentration set to a constant at all nodes. CARD E.l {F10.5) 1-10 : CINT Initial concentration - kg/n? ^CONC = 2 Read in initial concentration for each node. CARD E.l et.seq.(4(I10,F10.5)) 1-10 : IT{J) Node number 11-20 : TEMP{J) Initial concentration - kg/m^ Reading stops for IT(J) < 0 ^CONC = 3 Compute initial concentrations at each node using user supplied model in Subroutine CONCIC. For NOPT = 3 (Sediment Problems) : SET F INITIAL BED ELEVATIONS If IELEV+0, the initial bed elevation, with respect to some datum, at each node is read in. Read from file unit INE CARD F.l et.seq.(8F10.5) 1-10 : ELEV(I) Bed elevation for node I - m I = 1,...,NP For NOPT = 3: SET G ORIGINAL SETTLED BED PROFILE Read in only if IBED is not zero. Otherwise the default bed condition will be a clean bed. CARD G.l et.seq.(3I5,F10.5) for each element Element number Number of layers of original settled bed for element NN If NM=0, bed properties are read in for each element. If NM=i=0, constant values are read in and used for all elements. Average density of pore water in original bed - kg/m3 CARD G.2 et.seq.(2F10.5) Shear strength and dry sediment density for original settled bed layers. NLA+1 1-5 : NN 6-10 ■ NLA 11-15 NM 16-25 : GWA 318 pairs of values are read in starting at the top layer and proceeding downward. The first values are for the top of the original bed. 1-10 : SSTO(NN,L) Bed shear strength - N/m^ 11-20 : GADO(NN,L) Dry sediment density - kg/m^ For I = 1,...,NLA+1 CARD G.3 et.seq.(2E10.3) Thickness and value of M for each settled bed layer. 1-10 : THICKO(NN,I) Thickness of Ith_ layer - m 11-20 : EROCON{NN,I) M value for I th layer - kg/m^/s For I = 1,...,NLA Note: CARDS G.l, G.2 and G.3' are repeated for NM=1,...,NE when NM=0. When NM*0, these cards are read in only once. If stationary suspension is present on top of original settled bed, set NN=-10 at the end of the above set (i.e. after CARDS G.l, G.2 and G.3). For NN=-10, read the following cards. CARD G.4 et.seq.(I10,F10.5) for each element 1-10 : IT(J) Element number 11-20 : TEMP(J) Dry mass per unit area of (soft unconsolidated sediment) on top of settled bed - kg/m . Reading stops when IT(J) < 0 SET H INITIAL DEPTHS OF FLOW Depths of flow at each node are read in depending on the value of IDEP. Read from file unit INH IDEP = 0 All depths set to 1.0 m by default. IDEP = 1 All nodal depths set to constant. CARD H.l (Flo. 5) Constant value of depth. 1-10 : CDEP Depth of flow - m IDEP = 2 Read node point depths from file INH. CARD H.2 (4(110, FIO. 5)) 1-10 : IT(J) Node number 11-20 : TEMP(J) Depth of flow - m Stops reading if IT(J) < 0 319 IDEP = 3 Compute depths from user supplied procedure in Subroutine DEPTH. SET I INITIAL VELOCITY FIELD The horizontal velocity components at each node must be specified. The value of IVEL determines type of input. This input only for unsteady problems. All reads are from file number INF. IVEL = 1 Velocities are set to constant values. Read from file unit INF CARD I.l (2F10.5) 1-10 : CONXY Velocity ' component in the x-di recti on - m/s 11-20 : CONYV Velocity component in the y-direction - m/s IVEL = 2 Each nodal velocity component read in. CARD 1.2 (4(2F10.5)) Must be read in order for all NP nodes. 1-10 : XVEL(J,1) Velocity component at node J in the x- di recti on - m/s 11-20 : XVEL{J,2) Velocity component at node J in the y- di recti on - m/s IVEL = 3 User supplied procedure in Subroutine VEL is used to calculate nodal velocities. SET J INITIAL DISPERSION COEFFICIENTS The form of input is set by the value of IDIF, IDIF = 1 Dispersion coefficients are set to constant values. CARD J.l (2F10.5) 1-10 : CDIFL Longitidinal dispersion coefficient - m^/s 11-20 : CDIFT Transverse dispersion coefficient - m^/s IDIE = 2 Dispersion coefficients are read in node by node. CARD J. 2 et.seq.(3(I5,2F10.5) 1-5 : IT(J) Node number 6-15 : TEMP(1,J) Longitudinal dispersion coefficient - m2/s 16-25 : TEMP(2,J) Transverse dispersion coefficient - ".2/s m' 320 Reading stops for IT(J) < 0 IDIF = 3 Dispersion coefficients D^^^, D^^ , 0^^^ and D computed analytically using dispersion algorithm. For NOPT = 3 : SET K INITIAL SETTLING VELOCITIES The initial settling velocities at each node point must be read in. The form of input is determined by the value of I SET. Read from file unit ING ISET = 1 All settling velocities are set to constant. Card K.l {F10.5) 1-10 : CVSX Settling velocity - m/s ISET = 2 CARD K.2 et.seq.(4(I10,F10.5)) 1-10 : IT(J) Node number 11-20 : TEMP(J) Settling velocity - m/s Stops reading if IT(J) < 0 ISET = 3 Settling velocity model, for which parameters were read in SET D, is used to compute each nodal settling velocity. SET L BOUNDARY CONDITIONS For any problem, concentration boundary conditions must be specified at least at one node. At all external boundaries that have no concentration specified, the normal diffusive flux is defaulted to zero. Type of input is determined by value of IN8C. File number for input is INB. INBC = 1 Read node number and specified boundary condition from cards. CARD L.l (3(110, FIO. 5)) 1-10 : IT(J) Node number 11-20 : TEMP(J) Specified concentration - kg/m^ For J=1,...,NP INBC = 2 Concentration computed according to user supplied procedure in Subroutine CONCBC 321 I NBC = 3 Read node number and specified boundary condition from tape file 1MB. CARD L.2 MFIX(J) Equal to 1 for boundary node; 0 for all other nodes SPEC(J,1) Specified concentration at node J - kg/m"^ For J=1,...,NP SET M NEW SALINITIES Type of input is determined by value of ISALC{J), for J=2, . . . ,NTTS. ISALC(J)=1 CARD M.l (F10.5) 1-20 : SVI Constant salinity value read in - ppt ISALC(J)=2 File number for input - INSS CARD M.2 (3(110, FIO. 5)) Salinities for specified nodes 1-10 : IT(J) Node number 11-20 : TEMP (J) New salinity at node J - ppt Reading stops where IT(J) < 0. ISALC(J)=3 File number for input - INSS CARD M.3 (7F10.5) New salinities for all nodes. 1-10 : SAL(J) Salinity at Jth node for J=1,...,NP DYNAMIC INPUT The same subroutines that read initial values are used to read changes in these values during a dynamic run. The input code arrays in SET C tell the program if any new values should be read in at each time step. Note that the starting time is time step 1. The order of reading each set of data is given below. If the code is zero, no input is required. If it is non-zero, the value of the code will determine the type of input. 322 DESCRIPTION CODE ARRAY INPUT CARD SET Salinities ISALC(J) Concentration B.C. IFF(J,1) Depths of flow IDEPC(J) Velocities IVCOD(J) Dispersion coefficients IDIF(J) Settling velocities ISVS(J) SET M SET L SET SET SET SET I F E J only for sediment problems APPENDIX D DATA COLLECTION AND ANALYSIS PROGRAMS D.l. Field Data Collection Program A field data collection program for a sedimentation study should consist of four principal components: 1) hydrographic survey, 2) sediment sampling, 3) measurement of suspended sediment concentration, water temperature and salinity and 4) determination of sediment settling velocity. The collection program required for modeling the hydrodynamic regime in an estuary is rather well known, and will not be addressed. Hydrographic Survey At least two sonar fathometers and a gamma-ray transmission densitometer should be used to measure the depths in the water body to be modelled. First it is recommended that the entire water body be surveyed simultaneously using, for example, a 30-kHz and a 200-kHz fathometer. The Kelvin Hughes Division of Smiths Industries Ltd. (U.K.) manufactures a 30-kHz fathometer (model MS48) that has an approximate maximum range and resolution of 1000 m and 10 mm on the 0-20 range, respectively. Raytheon (USA) manufacturers a 200-kHz fathometer with an approximate maximum range and resolution of 120 m and 0.5% of depth, respectively. The 200-kHz fathometer has a maximum range that is approximately one order of magnitude less than that of the 30-kHz instrument because of the greater attenuation of the high frequencies in water. Raytheon does manufacture a dual 22.5-kHz and 200-kHz frequency fathometer. No specifications are available for this model. 323 324 Parker and Kirby (1977) reported that the sediment-water interfaces of stationary suspensions in the Severn Estuary. England were detected by a 200-kHz fathometer but not by a 30-kHz fathometer. In areas where stationary suspensions are determined to exist (by comparison of the 30- kHz and 200-kHz records), it is recommended that a gamma-ray transmission densitometer or a turbidity meter be used to supplement the depth record obtained with the fathometers. A fathometer alone may not be capable of detecting the surface of a stationary suspension for the following reasons: 1) The acoustic detection of a dense suspension depends on the gradient of the bulk density at the surface of the suspension and not on the magnitude of the density. In mobile suspensions and in newly formed stationary suspensions this density gradient is very small, and in most cases the surfaces of these suspensions will not be detectable. 2) As the stationary suspension undergoes consolidation, different parts (i.e. levels) of the suspension may become detectable to fathometers with different frequencies at different times, which makes the interpretation of such fathometer records a difficult and uncertain task (Parker and Kirby, 1977). A gamma-ray transmission densitometer obtains in situ measurements of the sediment bulk density profiles, and thus can be used, in addition, to determine the thickness of stationary suspensions and the location (i.e. vertical elevation with respect to geodetic datum) of the top of the settled bed, at which the bulk density is usually assumed to be 1300 kg/m . A static cone penetrometer directly measures penetration resistance and indirectly measures the shear strength of the sediment. The densitometer has to be calibrated at the beginning and end of survey operations to determine the relationship between the radiation count 325 rate and sediment density. Calibration is generally performed using liquids with different densities. The densitometer is penetrated and retracted in the sediment at a rate of approximately 2 to 3 mm/sec, during which the radiation count rate and probe penetration depth are continuously recorded. This system has been repeatedly used from both ships and submersibles and has the capability of measuring in situ bulk sediment densities up to 1800 kg/m3. operating in depths up to 3.6 km. and penetrating one to two meters in cohesive sediment suspensions (Hirst etal_^, 1975). Harwell, the United Kingdom Atomic Energy Research Establishment, manufacturers a transmission densitometer that has a vertical resolution of ± 1 cm with an accuracy of ± 2% in the bulk density range 1000-2000 kg/m^ (Parker etal_., 1975). There are three methods used to fix the boat position during a hydrographic survey: 1) optical methods. 2) electronic methods and 3) combined systems. Optical methods include double horizontal sextant angle resection, single horizontal sextant angle and transit line, and theodolite intersections from shore. Electronic methods include two megahertz systems, microwave systems and range and bearing systems, while combined systems use a theodolite to determine the bearing and a microwave system to determine the distance. A description of these three methods is given by Ingham (1975) and Dyer (1979). Sediment Sampling Using Corers Before the data collection period begins, at least two 10-12 cm diameter cores should be collected at each sampling station. There are four types of corers. These are the gravity-corer, piston-corer, vibracorer and box-corer. A gravity corer is lowered close to the ocean floor and a tripping mechanism is released so that the last part of the 326 descent is in free fall. The core barrel is supposed to penetrate the sediment, cutting out a cylinder of mud. The barrel is equipped with a plastic liner which can be slipped out. An orange peel core catcher is located between the nose piece and the liner to prevent the core from washing out when the corer is retrieved. In general, gravity corers have a barrel length of 1 to 2 m. A piston corer is released and free falls a known distance as soon as a tripping weight hanging a known distance below the corer senses the bottom. During penetration of the corer in the sediment the piston moves up the core liner. This action permits the hydrostatic pressure head of the water column to aid the corer in penetrating the sediment and removing the water from the barrel to reduce the resistance to the core as it enters the liner. Cores in excess of 20 m have been obtained from soft sediments. A vibrator mechanism is used to drive a vibracorer into sediment. In general, this type of corer is heavy and requires a large vessel for operation. However, a lighweight pneumatic corer has recently been designed and field tested at the U.S. Army Corps of Engineers Coastal Engineering Research Center (CERC) (Fuller and Meisburger, 1982). This system can be used from relatively small vessels for obtaining 5.0 cm diameter cores of unconsolidated sediments from 0.6 to 2.4 m in length. A box-corer is used to obtain large cores (up to 50 cm in height. 30 cm in length and 20 cm in width) of sediment when it is essential that the internal fabric of the sediment not be disturbed. Box-corers consist of an open steel box that is driven vertically into the sediment until the top of the box rests on the surface of the sediment. The open end of the box is covered by a blade which cuts through the sediment. 327 This type of corer has been used to obtain excellent cores of estuarine sediment (Dyer 1979). Measurement of Suspen ded Concentration. Salinity and Temperature The first item that must be considered is the time period over which data will be collected for eventual use in the model. This will be contingent upon the desired results from the modeling effort. In tidal water bodies, data should preferably be collected over a minimum of 15 hours (assuming the tide is semi-diurnal) over three different tidal cycles: spring, mean and neap. It would be more desirable to have the data collection period span at least one week, starting, for example, on a spring tide and finishing at the subsequent neap tide. The next consideration is the number of data sampling stations and where they should be located in order to adequately monitor the spatial variations of the concentration of suspended sediment. Stations must be located at all exterior water boundaries (cross-sections) of the estuarial system to be modeled. The width of the boundary cross-section and the lateral variability of the depth should be considered when deciding upon the minimum number of stations to be located laterally across such a boundary. For example, stations would definitely be located at predominant features such as navigation channels. Additional stations must be located at all interior confluences and bifurcations, and at as many other interior locations as possible. It is recognized that the length of the data collection period and the number of stations are often less than desired due to economic and logistical considerations. At each station the location of the top of the sediment bed with respect to a geodetic or tidal datum must be determined using the 328 previously described surveying methods. It is recommended the water temperature, electrical conductivity (or salinity) and concentration of suspended sediment be measured at least once every one-half hour for the duration of the collection period at each of the sampling stations. These measurements should preferably be made at a minimum of three depths over the vertical: one-half meter below the water surface, mid- depth and one-half meter above the bottom (i.e. top of sediment bed). For locations where the water depth is greater than about 3 to 4 m, measurements should be made at additional depths over the vertical. Both the measurement and analysis of water temperature and electrical conductivity data are discussed by Dyer (1979). A description of various filtration procedures for determining the suspension concentration gravi metrically is given by Dyer (1979). There are three general methods used to measure the suspension concentration: 1) water sampling, 2) optical methods and 3) gamma-ray densitometer measurements. Water bottles and shipboard pumps are the two most common water sampling devices. The NIO bottle has capacities from 1.25 to 7.1 liters. Other water bottles, such as the Van Dorn bottle, have capacities of up to 10 liters or more. The NIO bottle consists of a PVC tube open at both ends with hemispherical bings on spring loaded arms which close each end. Most water bottles are closed by dropping a brass messenger down the support wire (Dyer, 1979). Shipboard pumps are used to pump water samples up to the vessel through an intake tube mounted on an instrument package. It is recommended that in situ separation of the water and sediment be performed on the vessel using the filter method (van Rijn, 1979). 329 Instruments for optical determination of the concentration of suspended sediment include the transmissometer, nephelometer and the Secchi disc. Nephelometers are not very practical for use in estuaries since they are sensitive to very low concentrations only. Secchi discs can be used to estimate surface values only. Transmissometers, or electro-optical turbidity meters, have been used successfully to measure vertical turbidity profiles in, among others, the Severn, Maas, James and Rappahannock estuaries and in Upper Chesapeake Bay (Kirby and Parker, 1977; Nichols ^al_., 1979). These meters can be used to detect both mobile and stationary suspensions as their operating range, in general, is 0.25-25 g/1 . They have a rapid response time (100 Hz) which allows profiles in 30 m depths to be taken in 15-20 seconds. Partech (U.K.) is one manufacturer of optical turbidity meters. Both Kirby and Parker (1977) and Nichols etaj_. (1979) used instrument arrays on which were mounted a gamma-ray transmission densitometer, at least one electro-optical turbidity meter, an electromagnetic current meter, a pressure transducer and a water temperature and conductivity probe. In general, the optical turbidity meters would be used to record the concentration profile for suspensions up to 25 g/1 and the transmission densitometer used for denser suspensions. Determination of Sediment Settling Velocity An appropriate method to measure settling velocities is by using an instrument similar to the sampling tube developed by Owen (1971), in which undisturbed samples of suspended sediments are collected in situ in their natural state. The settling velocities of the aggregates are 330 determined immediately thereafter through use of a bottom withdrawal sedimentation test. Allersma (1980) gives a detailed description of an I'n situ suspended sediment sampler. D.2 Laboratory Sediment Testing Program It is recommended that the following physiochemical sediment and fluid properties be determined using the collected sediment cores. Properties of Undisturbed Sediment Cores The gamma-ray densitometer may be used to determine the bulk sediment density profile in the undisturbed cores still in the liner tubes as soon after the cores are obtained as possible. A description of this procedure is given by Whitmarsh (1971) and Kirby and Parker (1974). If this instrument is not available, the freeze-drying procedure used by Parchure (1980) and Dixit (1982) or the pumping method used by Thorn and Parsons (1977) may be used to determine the bulk density profile. The pumping method consists of removing by suction a thin layer, e.g. 3 cm, from the top of the core. This procedure is repeated, layer by layer, and each layer is analyzed to determine the mean bulk density. Properties of Original Settled Bed The bulk density and bed shear strength profiles and the erosion rate constant for each layer need to be determined for the cores. The number of layers and the thickness of each are determined from the nature of the bed shear strength profile. The erosion rate constant for each layer and the shear strength profile can be determined, for example, in the rotating cylinder erodibility testing apparatus described by Sargunam et al_. (1973). In order to use this apparatus. 331 the core sample must be trimmed. The portion of each core that is sufficiently consolidated such that it can be trimmed and tested in the erosion apparatus is defined to be the settled bed. The thickness of this portion defines the location of the top of the settled bed. Soft, unconsolidated portions of each core are assumed to be new deposits. Properties of New Deposits For cores with soft, unconsolidated or partially consolidated sediment on top of the settled portion, the following method may be used to estimate the erosional, depositional and consolidation characteristics of such new deposits. The new deposit samples from the cores at all the stations should be mixed and subjected to laboratory erosion, deposition and consolidation tests (described by Parchure (1980), Mehta and Partheniades (1973). in Sections 3.4 and 3.6 and below) to determine: the settling velocity as a function of suspension concentration and salinity; the minimum and maximum depositional shear stresses -r,^^^.^ and -z^^^^; variation of tgg and with the bed shear stress, -v^; the number of characteristic stationary suspension layers, and the thickness, dry sediment density and shear strength of each layer; the number of characteristic partially consolidated new deposit layers, and the thickness, dry sediment density, shear strength and the resuspension parameters and a of each layer; the variation of L and Tdc„ with C^; variation of p(z) with T^^; variation of the bed shear strength, t^, with p. The variation of the bed density and shear strength profiles with salinity can be determined by performing the erosion tests at several salinities between 0 and 35 ppt. A brief description of laboratory tests which can be conducted in order to determine the above mentioned consolidation parameters is given next. 332 Laboratory tests to determine the consolidation characteristics of a cohesive sediment bed involve the measurement of the bed density profile. Various methods have been used for this purpose. Been and Sills (1981) measured the density profile of a clayey soil using a non- destructive X-ray technique. Methods which involve the destruction of the soil include, among others, the freeze-drying procedure used by Parchure (1980) and Dixit (1982), the pumping, or layer-by-layer sampling method used by Thorn and Parsons (1977), and the use of specially designed apparatuses (Parchure, 1980). The latter consists of a 183 cm high, 30 cm diameter PVC cylinder, a bottom plate, and ten 1.27 cm diameter plastic tubes ranging from 1.27 to 12.7 cm in height glued to the bottom plate, concentric to the PVC cylinder. These cylinders are filled with a sediment suspension of known concentration, and the sediment is allowed to settle under quiescent conditions for a specified consolidation time. Following the procedure described by Parchure (1980), the bulk density profile can be determined. The following parameters should be varied systematically in the laboratory tests in order to determine their effect on the rate of consolidation: (a) Consolidation time - it is recommended that the time allowed for the bed to consolidate before the density profile is measured be varied logarithmically from 0 to 720 hours (1 month). (b) Initial conditions - the initial suspension concentration, which determines the thickness and density of the initial bed. (c) Salinity - the salinity of the water should be varied from 0 (tap water) to 35 ppt. 333 (d) Overburden - it is important that the effect of discretized to continuous additions of varying amounts of sediment (overburden) on top of the initial bed be determined in order to evaluate the effect of such overburden pressures on the consolidation rate of the lower bed layers. The relationship between p and -c^ needs to be determined as well for the collected sediment samples. Both the bed shear strength and density profiles may be determined using the methodology described by Mehta et al. (1982a). These profiles can then be used to establish an empirical relationship between and p. Fluid Composition The pH, total salt concentration, and concentrations of ions such as Na"*", Ca^"^, Mg^"^, K"^, Fe^"^ and CI" should be determined for both the pore fluid in the consolidated bed portion of one core and a sample of the suspending fluid. Composition and Cation Exchange Capacity of the Sediment The sediment contained in the consolidated bed portion of one core from each collection station should be thoroughly mixed so that a spatially homogeneous sample is obtained. A standard hydrometer analysis must be conducted on each so-prepared sample to determine the sediment particle size distribution and thereby the percentage by weight of clay, silt and fine to coarse sand in each sample. In preparing the samples for this analysis, the sediment must not be initially air-dried (to obtain the dry weight of the material used in the test), as it has been found that dried sediment will not completely redisperse when the dispersing agent is added (Krone, 1962). For this reason, the total dry weight of the sample must be obtained after the test by evaporating off 334 all the water in an oven set at approximately 50° C. 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J. Nihoul, Editor, Elsevier, Amsterdam, 1978, pp. 207-216. Znidarcic„ D., "Laboratory Determination of Consolidation Properties for Cohesive Soil," Ph.D. Dissertation . University of Colorado, Boulder, Colorado, 1982. BIOGRAPHICAL SKETCH Earl Joseph Hayter was born November 19, 1954, in Coral Gables, Florida. At the age of 9 his family moved to Apopka, Florida, where he completed his primary and secondary education and graduated from Lyman High School in 1972. He attended Florida Technological University from September 1972 to June 1974. In September 1974 he transferred to Florida Institute of Technology and majored in physical oceanography. He graduated from F.I.T. in June 1975 with a Bachelor of Science degree in oceanography. In September 1976 he enrolled as a graduate student in the Coastal and Oceanographic Engineering Department at the University of Florida. He graduated from the University of Florida in March 1979 with a Master of Science degree in coastal and oceanography engineering. From September 1977 to May 1979 he worked for Suboceanic Consultants, Inc., in Naples, Florida. He reentered graduate school at the University of Florida in June 1979 in the Department of Civil Engineering to work toward the Doctor of Philosophy degree. On June 21, 1980, he married Janet Griffen. He worked as a graduate research assistant in the Coastal and Oceanographic Engineering Department during the periods September 1976 - September 1977 and June 1979 - September 1981. From October 1981 to September 1983, he was supported by the Water Resources Division of the U.S. Geological Survey under their thesis support program. 349 I certify that I have read this study and that in my opinion 1t conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Bent A. Christensen, C hai rman Professor of Civil Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy. /A sh^*W^J. Mehta, C o"- C h a i r ma h Associate Professor of Coastal and Oceanographic Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Barry ^ Benedict Professor of Civil Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy. WAjL^ ex rQ.r^!l^^4^ /J^mes ~. Eades Associate Professor of Geology Tin's dissertation was submitted to the Dean of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1983 Dean, College of Engineering Dean, Graduate School